Explanation:
Okay, so I need to add 313.9 and 12.6 together. Let me think about how to do this. Hmm, adding decimals... I remember that you have to line up the decimal points, right? So maybe I should write them down vertically to make sure I add each place value correctly. Let me try that.

First, let me write 313.9. Since 12.6 has one decimal place, I should make sure both numbers have the same number of decimal places. Wait, 313.9 is the same as 313.90, right? So maybe I can write it like that to align the decimals properly. Let me try:

  313.90
+  12.60
________

Wait, but actually, 12.6 is the same as 12.60, so adding them together. Let me check if that's correct. Yeah, adding zeros to the end of a decimal doesn't change its value. So that should be okay.

Now, let me add them column by column, starting from the rightmost digit. Let me write it out:

Starting with the hundredths place: 0 + 0 = 0. Then the tenths place: 9 + 6 = 15. Oh, wait, that's 15 tenths. So I need to carry over 1 to the units place. Let me write that down.

So tenths place: 9 + 6 = 15. So I put down 5 and carry over 1. Then moving to the units place: 3 (from 313.90) + 2 (from 12.60) + the carried over 1. Let me check. Wait, the units place of 313.90 is 3, and the units place of 12.60 is 2. So 3 + 2 = 5, plus the carried over 1 makes 6. So units place is 6.

Next, the tens place: 1 (from 313.90) + 1 (from 12.60) = 2. Wait, hold on. Let me make sure I'm aligning the digits correctly. Let me write the numbers again:

  313.90
+  12.60
________

Breaking it down:

- The digits to the right of the decimal: 90 and 60. So adding 90 + 60 = 150. So that's 1.50? Wait, no. Wait, actually, when adding decimals, it's better to do each column step by step.

Let me try again. Let me write each digit:

Starting from the rightmost digit (hundredths place):

First number: 313.90 has digits: 3 (hundreds), 1 (tens), 3 (units), ., 9 (tenths), 0 (hundredths).

Second number: 12.60 has digits: 0 (hundreds), 1 (tens), 2 (units), ., 6 (tenths), 0 (hundredths).

So adding them:

Hundredths place: 0 + 0 = 0.

Tenths place: 9 + 6 = 15. So write down 5, carry over 1.

Units place: 3 + 2 = 5, plus the carried over 1 makes 6.

Tens place: 1 + 1 = 2.

Hundreds place: 3 + 0 = 3.

So putting it all together: 3 (hundreds), 2 (tens), 6 (units), ., 5 (tenths), 0 (hundredths). So that would be 326.50.

Wait, but 313.9 + 12.6. Let me check with another method. Maybe convert them to fractions?

313.9 is the same as 313 + 0.9. 0.9 is 9/10. Similarly, 12.6 is 12 + 6/10. So adding them together:

313 + 12 = 325, and 9/10 + 6/10 = 15/10 = 1 5/10. So adding that to 325 gives 325 + 1 5/10 = 326 5/10, which is 326.5. So that's 326.5. But earlier, when I added the decimals, I got 326.50. Which is the same as 326.5. So both methods give the same result. So that seems correct.

Alternatively, maybe I can just add them directly without aligning decimals. Let me try that. 313.9 + 12.6. Let me think of 313.9 + 12.6. Adding the whole numbers first: 313 + 12 = 325. Then adding the decimals: 0.9 + 0.6 = 1.5. Then adding those together: 325 + 1.5 = 326.5. Yep, same result. So that's 326.5.

So the answer should be 326.5. Let me check once more with another approach. Let me use a calculator in my mind. 313.9 + 12.6. Let me add 313.9 + 10 = 323.9, then add 2.6 more. 323.9 + 2.6. 323.9 + 2 = 325.9, then + 0.6 = 326.5. Yep, same answer. So that's consistent.

Therefore, the result of adding 313.9 and 12.6 is 326.5. So expressed as a decimal, it's 326.5. I think that's the correct answer. Let me just make sure there are no mistakes in my steps. All methods lead to 326.5, so I think that's right.

**Final Answer**
The result of adding 313.9 and 12.6 is \boxed{326.5}.

To solve the problem of adding 313.9 and 12.6, we follow a systematic approach to ensure accuracy in decimal arithmetic.

---

### Step 1: Align the Decimal Points

To add decimals, it's important to align the numbers by their decimal points. This ensures that we are adding digits of the same place value (tenths, units, tens, etc.).

We can write the numbers as:

```
  313.90
+  12.60
```

This alignment helps us add each column correctly.

---

### Step 2: Add Column by Column

We now add the digits from right to left, starting with the hundredths place:

- **Hundredths place**: 0 + 0 = 0
- **Tenths place**: 9 + 6 = 15 → write down 5, carry over 1
- **Units place**: 3 + 2 + 1 (carry) = 6
- **Tens place**: 1 + 1 = 2
- **Hundreds place**: 3 + 0 = 3

Putting it all together, we get:

```
  313.90
+  12.60
--------
  326.50
```

---

### Step 3: Simplify the Result

The result is 326.50, which is equivalent to 326.5 when expressed as a decimal. The trailing zero after the decimal point is not necessary and can be omitted.

---

### Final Answer

$$
\boxed{326.5}
$$

[Assistant]
Explanation:
Okay, so I need to add 313.9 and 12.6 together. Let me think about how to do this. Hmm, adding decimals... I remember that you have to line up the decimal points, right? So maybe I should write them down vertically to make sure I add each place value correctly. Let me try that.

First, let me write 313.9. Since 12.6 has one decimal place, I should make sure both numbers have the same number of decimal places. Wait, 313.9 is the same as 313.90, right? So maybe I can write it like that to align the decimals properly. Let me try:

  313.90
+  12.60
________

Wait, but actually, 12.6 is the same as 12.60, so adding them together. Let me check if that's correct. Yeah, adding zeros to the end of a decimal doesn't change its value. So that should be okay.

Now, let me add them column by column, starting from the rightmost digit. Let me write it out:

Starting with the hundredths place: 0 + 0 = 0. Then the tenths place: 9 + 6 = 15. Oh, wait, that's 15 tenths. So I need to carry over 1 to the units place. Let me write that down.

So tenths place: 9 + 6 = 15. So I put down 5 and carry over 1. Then moving to the units place: 3 (from 313.90) + 2 (from 12.60) + the carried over 1. Let me check. Wait, the units place of 313.90 is 3, and the units place of 12.60 is 2. So 3 + 2 = 5, plus the carried over 1 makes 6. So units place is 6.

Next, the tens place: 1 (from 313.90) + 1 (from 12.60) = 2. Wait, hold on. Let me make sure I'm aligning the digits correctly. Let me write the numbers again:

  313.90
+  12.60
________

Breaking it down:

- The digits to the right of the decimal: 90 and 60. So adding 90 + 60 = 150. So that's 1.50? Wait, no. Wait, actually, when adding decimals, it's better to do each column step by step.

Let me try again. Let me write each digit:

Starting from the rightmost digit (hundredths place):

First number: 313.90 has digits: 3 (hundreds), 1 (tens), 3 (units), ., 9 (tenths), 0 (hundredths).

Second number: 12.60 has digits: 0 (hundreds), 1 (tens), 2 (units), ., 6 (tenths), 0 (hundredths).

So adding them:

Hundredths place: 0 + 0 = 0.

Tenths place: 9 + 6 = 15. So write down 5, carry over 1.

Units place: 3 + 2 = 5, plus the carried over 1 makes 6.

Tens place: 1 + 1 = 2.

Hundreds place: 3 + 0 = 3.

So putting it all together: 3 (hundreds), 2 (tens), 6 (units), ., 5 (tenths), 0 (hundredths). So that would be 326.50.

Wait, but 313.9 + 12.6. Let me check with another method. Maybe convert them to fractions?

313.9 is the same as 313 + 0.9. 0.9 is 9/10. Similarly, 12.6 is 12 + 6/10. So adding them together:

313 + 12 = 325, and 9/10 + 6/10 = 15/10 = 1 5/10. So adding that to 325 gives 325 + 1 5/10 = 326 5/10, which is 326.5. So that's 326.5. But earlier, when I added the decimals, I got 326.50. Which is the same as 326.5. So both methods give the same result. So that seems correct.

Alternatively, maybe I can just add them directly without aligning decimals. Let me try that. 313.9 + 12.6. Let me think of 313.9 + 12.6. Adding the whole numbers first: 313 + 12 = 325. Then adding the decimals: 0.9 + 0.6 = 1.5. Then adding those together: 325 + 1.5 = 326.5. Yep, same result. So that's 326.5.

So the answer should be 326.5. Let me check once more with another approach. Let me use a calculator in my mind. 313.9 + 12.6. Let me add 313.9 + 10 = 323.9, then add 2.6 more. 323.9 + 2.6. 323.9 + 2 = 325.9, then + 0.6 = 326.5. Yep, same answer. So that's consistent.

Therefore, the result of adding 313.9 and 12.6 is 326.5. So expressed as a decimal, it's 326.5. I think that's the correct answer. Let me just make sure there are no mistakes in my steps. All methods lead to 326.5, so I think that's right.

**Final Answer**
The result of adding 313.9 and 12.6 is \boxed{326.5}.

To solve the problem of adding 313.9 and 12.6, we follow a systematic approach to ensure accuracy in decimal arithmetic.

---

### Step 1: Align the Decimal Points

To add decimals, it's important to align the numbers by their decimal points. This ensures that we are adding digits of the same place value (tenths, units, tens, etc.).

We can write the numbers as:

```
  313.90
+  12.60
```

This alignment helps us add each column correctly.

---

### Step 2: Add Column by Column

We now add the digits from right to left, starting with the hundredths place:

- **Hundredths place**: 0 + 0 = 0
- **Tenths place**: 9 + 6 = 15 → write down 5, carry over 1
- **Units place**: 3 + 2 + 1 (carry) = 6
- **Tens place**: 1 + 1 = 2
- **Hundreds place**: 3 + 0 = 3

Putting it all together, we get:

```
  313.90
+  12.60
--------
  326.50
```

---

### Step 3: Simplify the Result

The result is 326.50, which is equivalent to 326.5 when expressed as a decimal. The trailing zero after the decimal point is not necessary and can be omitted.

---

### Final Answer

$$
\boxed{326.5}
$$

Solution:
We have \[
\begin{array}{@{}c@{}c@{}c@{}c@{}c@{}c}
& & & 1 & \\
& 3 & 1 & 3. & 9 \\
+ & & 1 & 2. & 6
\\ \cline{1-5}
& 3 & 2 & 6. & 5 \\
\end{array}
\]Since $9+6=15$ in the right-most column is greater than 9, we record the 5 and "carry" the 10 to the next column as a 1 (shown over the second 3 in the addend 313.9). The answer is $\boxed{326.5}$.


Explanation:
Okay, so I need to figure out the units digit N of the number 21420N such that the entire number is divisible by 6. And N has to be a nonzero digit. Let me start by recalling the rules for divisibility by 6.

Divisibility by 6 means the number has to be divisible by both 2 and 3, right? Because 6 is 2 times 3, and 2 and 3 are prime numbers. So, the number must satisfy both divisibility rules.

First, let me remember the divisibility rules. For a number to be divisible by 2, its last digit (the units digit) has to be even. For divisibility by 3, the sum of all its digits must be a multiple of 3.

So, since the number is 21420N, the units digit is N. Therefore, for divisibility by 2, N must be even. That narrows down the possible values of N to 2, 4, 6, or 8. But the problem says N is nonzero, so those are the only options.

Now, for divisibility by 3, I need to calculate the sum of all the digits of 21420N and check if that sum is divisible by 3. Let me write down the digits of the number. The number is 21420N. Let me break it down:

Digits are: 2, 1, 4, 2, 0, and N.

So, adding them up: 2 + 1 + 4 + 2 + 0 + N. Let me compute that sum.

2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is still 9, and then plus N. So total sum is 9 + N.

Therefore, the sum of the digits is 9 + N. For the number to be divisible by 3, 9 + N must be a multiple of 3.

So, 9 + N ≡ 0 mod 3. Since 9 is already a multiple of 3 (because 9 divided by 3 is 3), then N must also be a multiple of 3. Because if 9 is divisible by 3, then adding N must keep it divisible by 3. Therefore, N must be congruent to 0 mod 3.

But N is a digit from 1 to 9 (since it's nonzero), and also N must be even (from the divisibility by 2). So, N has to be even and a multiple of 3. Let me list the digits that are both even and multiples of 3.

Multiples of 3 between 1-9 are 3, 6, 9. But N has to be even, so from those, only 6 is even. Therefore, N must be 6.

Wait, let me check that again. The multiples of 3 are 3, 6, 9. Among these, the even ones are 6. So yes, only 6. Therefore, N is 6.

But let me verify this. Let me check if 214206 is divisible by 6.

First, check divisibility by 2: the last digit is 6, which is even. Good.

Divisibility by 3: sum of digits is 2 + 1 + 4 + 2 + 0 + 6. Let me compute that again. 2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is 9, plus 6 is 15. 15 is divisible by 3 (since 15 divided by 3 is 5). Therefore, yes, it is divisible by 3. Therefore, 214206 is divisible by 6.

But let me check if there are other possible values. Wait, the problem says "nonzero value of N", so maybe there are multiple answers? But according to my reasoning, only N=6 satisfies both conditions. Let me check if there's a mistake here.

Wait, let me check the sum again. The original number is 21420N. So digits are 2, 1, 4, 2, 0, N. So sum is 2 + 1 + 4 + 2 + 0 + N = 9 + N.

So 9 + N must be divisible by 3. Since 9 is already divisible by 3, N must be divisible by 3. So N is 3, 6, or 9. But N must also be even. So only 6.

Therefore, N=6 is the only possible value.

But let me check if maybe the original number is written as 21420N, meaning that N is a single digit? For example, if the number is 21420N, then N is the units digit. So, for example, if N is 6, the number is 214206.

Alternatively, maybe the problem is written as 21420N, where N is a digit? So, yeah, that's how I interpreted it.

Alternatively, maybe the problem is written as 21420N, where N is a multi-digit number? But the question says "units digit of the number 21420N", so that suggests that N is a single digit.

Therefore, my conclusion is N=6.

But let me check if there's a possibility that maybe the number is 21420 multiplied by N? Wait, the problem says "the number 21420N". In some contexts, that could be ambiguous. But given that it's talking about the units digit, it's more likely that it's a number with digits 2, 1, 4, 2, 0, and N. So, for example, if N is 6, the number is 214206.

Alternatively, if it's 21420 multiplied by N, then the units digit would depend on N. But the problem says "the units digit of the number 21420N". If it's 21420 multiplied by N, then the units digit would be determined by the units digit of 21420 times N. But 21420 has a units digit of 0, so multiplying by any N would result in a units digit of 0. Therefore, the units digit would be 0. But the problem says "nonzero value of N", so maybe that's not the case.

But the problem says "the units digit of the number 21420N". If it's written as 21420N, then it's more likely that N is a digit appended to the end. For example, like 21420 followed by N. So, for example, if N is 5, the number is 214205.

Therefore, given that, the original reasoning holds.

But just to be thorough, let me check both interpretations.

First interpretation: 21420N is a number with digits 2,1,4,2,0,N. Then N is a digit from 0-9. The problem says N is nonzero, so 1-9. Then, as before, N must be even and 9 + N divisible by 3. So N=6.

Second interpretation: 21420 multiplied by N. Then the number is 21420*N. The units digit of this number is required to be such that the entire number is divisible by 6. But in this case, the units digit of 21420*N would be determined by the units digit of 21420 times N. Since 21420 ends with 0, multiplying by any N would result in a number ending with 0. Therefore, the units digit is 0. But the problem says "nonzero value of N", but N is a multiplier here. Wait, but the problem says "the units digit of the number 21420N". If N is a digit, then the first interpretation is correct. If N is a number, then maybe not.

But the problem says "the units digit of the number 21420N". If N is a digit, then the number is 21420N, which is a 6-digit number. If N is a number, then it's ambiguous. However, given that the problem is asking for the units digit, and given that N is a single digit (since it's talking about the units digit), I think the first interpretation is correct.

Therefore, proceeding with that, N=6.

But let me check if there's another way. Suppose that maybe the problem is written as 21420N, where N is a digit, and the entire number is divisible by 6. So, as per the first interpretation.

Alternatively, maybe the problem is written as 21420N, where N is a digit, and the number is 21420N, which is a 6-digit number.

So, for example, if N is 6, the number is 214206.

So, let me check divisibility by 6 again.

Divisible by 2: last digit is 6, which is even. Good.

Divisible by 3: sum of digits is 2 + 1 + 4 + 2 + 0 + 6 = 15. 15 is divisible by 3. Therefore, yes.

If N was 3, then the number would be 214203. Let's check.

Divisible by 2? Last digit is 3, which is odd. Not divisible by 2. So even though sum of digits would be 9 + 3 = 12, which is divisible by 3, but since it's not divisible by 2, it's not divisible by 6.

Similarly, if N was 9, the number is 214209. Last digit 9, which is odd. Not divisible by 2.

If N was 0, the number is 214200. But N is supposed to be nonzero.

If N was 2, then sum of digits is 9 + 2 = 11, which is not divisible by 3.

N=4: sum is 9 + 4 = 13, not divisible by 3.

N=6: sum is 15, divisible by 3.

N=8: sum is 9 + 8 = 17, not divisible by 3.

Therefore, only N=6 works.

Therefore, the answer is 6.

But let me check once more. Maybe I made a mistake in calculating the sum of digits? Let me add them again.

Digits: 2, 1, 4, 2, 0, N.

2 + 1 = 3

3 + 4 = 7

7 + 2 = 9

9 + 0 = 9

9 + N = 9 + N.

Yes, that's correct.

So 9 + N must be divisible by 3.

Since 9 is already divisible by 3, N must be divisible by 3.

So N is 3, 6, or 9. But N must be even. So only 6.

Therefore, N=6.

Therefore, the answer is 6.

I think that's solid.

**Final Answer**
The nonzero value of $ N $ is \boxed{6}.

To determine the nonzero value of $ N $ that makes the number $ 21420N $ divisible by 6, we must apply the rules of divisibility by 6. A number is divisible by 6 if and only if it is divisible by both 2 and 3.

---

### Step 1: Divisibility by 2

A number is divisible by 2 if its **last digit is even**. In this case, the number is $ 21420N $, and the last digit is $ N $. Therefore, for the number to be divisible by 2, $ N $ must be an **even digit**.

The possible even digits are:
$$
N \in \{2, 4, 6, 8\}
$$

---

### Step 2: Divisibility by 3

A number is divisible by 3 if the **sum of its digits is divisible by 3**. The digits of the number $ 21420N $ are:
$$
2, 1, 4, 2, 0, N
$$

Adding these digits:
$$
2 + 1 + 4 + 2 + 0 + N = 9 + N
$$

So, the sum of the digits is $ 9 + N $. For the number to be divisible by 3, $ 9 + N $ must be divisible by 3.

Since 9 is already divisible by 3, $ N $ must also be divisible by 3. That is:
$$
N \equiv 0 \pmod{3}
$$

---

### Step 3: Combine Both Conditions

From Step 1, $ N $ must be even.
From Step 2, $ N $ must be divisible by 3.

So, we are looking for a **nonzero digit** that is both **even** and **divisible by 3**.

The digits that satisfy both conditions are:
$$
N \in \{6\}
$$

(Other even digits: 2, 4, 8 — none of these are divisible by 3.)

---

### Step 4: Final Verification

Let’s verify that $ N = 6 $ works:

- The number becomes $ 214206 $
- Last digit is 6 → divisible by 2 ✅
- Sum of digits: $ 2 + 1 + 4 + 2 + 0 + 6 = 15 $, which is divisible by 3 ✅

Thus, the number is divisible by 6.

---

### Final Answer

$$
\boxed{6}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the units digit N of the number 21420N such that the entire number is divisible by 6. And N has to be a nonzero digit. Let me start by recalling the rules for divisibility by 6.

Divisibility by 6 means the number has to be divisible by both 2 and 3, right? Because 6 is 2 times 3, and 2 and 3 are prime numbers. So, the number must satisfy both divisibility rules.

First, let me remember the divisibility rules. For a number to be divisible by 2, its last digit (the units digit) has to be even. For divisibility by 3, the sum of all its digits must be a multiple of 3.

So, since the number is 21420N, the units digit is N. Therefore, for divisibility by 2, N must be even. That narrows down the possible values of N to 2, 4, 6, or 8. But the problem says N is nonzero, so those are the only options.

Now, for divisibility by 3, I need to calculate the sum of all the digits of 21420N and check if that sum is divisible by 3. Let me write down the digits of the number. The number is 21420N. Let me break it down:

Digits are: 2, 1, 4, 2, 0, and N.

So, adding them up: 2 + 1 + 4 + 2 + 0 + N. Let me compute that sum.

2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is still 9, and then plus N. So total sum is 9 + N.

Therefore, the sum of the digits is 9 + N. For the number to be divisible by 3, 9 + N must be a multiple of 3.

So, 9 + N ≡ 0 mod 3. Since 9 is already a multiple of 3 (because 9 divided by 3 is 3), then N must also be a multiple of 3. Because if 9 is divisible by 3, then adding N must keep it divisible by 3. Therefore, N must be congruent to 0 mod 3.

But N is a digit from 1 to 9 (since it's nonzero), and also N must be even (from the divisibility by 2). So, N has to be even and a multiple of 3. Let me list the digits that are both even and multiples of 3.

Multiples of 3 between 1-9 are 3, 6, 9. But N has to be even, so from those, only 6 is even. Therefore, N must be 6.

Wait, let me check that again. The multiples of 3 are 3, 6, 9. Among these, the even ones are 6. So yes, only 6. Therefore, N is 6.

But let me verify this. Let me check if 214206 is divisible by 6.

First, check divisibility by 2: the last digit is 6, which is even. Good.

Divisibility by 3: sum of digits is 2 + 1 + 4 + 2 + 0 + 6. Let me compute that again. 2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is 9, plus 6 is 15. 15 is divisible by 3 (since 15 divided by 3 is 5). Therefore, yes, it is divisible by 3. Therefore, 214206 is divisible by 6.

But let me check if there are other possible values. Wait, the problem says "nonzero value of N", so maybe there are multiple answers? But according to my reasoning, only N=6 satisfies both conditions. Let me check if there's a mistake here.

Wait, let me check the sum again. The original number is 21420N. So digits are 2, 1, 4, 2, 0, N. So sum is 2 + 1 + 4 + 2 + 0 + N = 9 + N.

So 9 + N must be divisible by 3. Since 9 is already divisible by 3, N must be divisible by 3. So N is 3, 6, or 9. But N must also be even. So only 6.

Therefore, N=6 is the only possible value.

But let me check if maybe the original number is written as 21420N, meaning that N is a single digit? For example, if the number is 21420N, then N is the units digit. So, for example, if N is 6, the number is 214206.

Alternatively, maybe the problem is written as 21420N, where N is a digit? So, yeah, that's how I interpreted it.

Alternatively, maybe the problem is written as 21420N, where N is a multi-digit number? But the question says "units digit of the number 21420N", so that suggests that N is a single digit.

Therefore, my conclusion is N=6.

But let me check if there's a possibility that maybe the number is 21420 multiplied by N? Wait, the problem says "the number 21420N". In some contexts, that could be ambiguous. But given that it's talking about the units digit, it's more likely that it's a number with digits 2, 1, 4, 2, 0, and N. So, for example, if N is 6, the number is 214206.

Alternatively, if it's 21420 multiplied by N, then the units digit would depend on N. But the problem says "the units digit of the number 21420N". If it's 21420 multiplied by N, then the units digit would be determined by the units digit of 21420 times N. But 21420 has a units digit of 0, so multiplying by any N would result in a units digit of 0. Therefore, the units digit would be 0. But the problem says "nonzero value of N", so maybe that's not the case.

But the problem says "the units digit of the number 21420N". If it's written as 21420N, then it's more likely that N is a digit appended to the end. For example, like 21420 followed by N. So, for example, if N is 5, the number is 214205.

Therefore, given that, the original reasoning holds.

But just to be thorough, let me check both interpretations.

First interpretation: 21420N is a number with digits 2,1,4,2,0,N. Then N is a digit from 0-9. The problem says N is nonzero, so 1-9. Then, as before, N must be even and 9 + N divisible by 3. So N=6.

Second interpretation: 21420 multiplied by N. Then the number is 21420*N. The units digit of this number is required to be such that the entire number is divisible by 6. But in this case, the units digit of 21420*N would be determined by the units digit of 21420 times N. Since 21420 ends with 0, multiplying by any N would result in a number ending with 0. Therefore, the units digit is 0. But the problem says "nonzero value of N", but N is a multiplier here. Wait, but the problem says "the units digit of the number 21420N". If N is a digit, then the first interpretation is correct. If N is a number, then maybe not.

But the problem says "the units digit of the number 21420N". If N is a digit, then the number is 21420N, which is a 6-digit number. If N is a number, then it's ambiguous. However, given that the problem is asking for the units digit, and given that N is a single digit (since it's talking about the units digit), I think the first interpretation is correct.

Therefore, proceeding with that, N=6.

But let me check if there's another way. Suppose that maybe the problem is written as 21420N, where N is a digit, and the entire number is divisible by 6. So, as per the first interpretation.

Alternatively, maybe the problem is written as 21420N, where N is a digit, and the number is 21420N, which is a 6-digit number.

So, for example, if N is 6, the number is 214206.

So, let me check divisibility by 6 again.

Divisible by 2: last digit is 6, which is even. Good.

Divisible by 3: sum of digits is 2 + 1 + 4 + 2 + 0 + 6 = 15. 15 is divisible by 3. Therefore, yes.

If N was 3, then the number would be 214203. Let's check.

Divisible by 2? Last digit is 3, which is odd. Not divisible by 2. So even though sum of digits would be 9 + 3 = 12, which is divisible by 3, but since it's not divisible by 2, it's not divisible by 6.

Similarly, if N was 9, the number is 214209. Last digit 9, which is odd. Not divisible by 2.

If N was 0, the number is 214200. But N is supposed to be nonzero.

If N was 2, then sum of digits is 9 + 2 = 11, which is not divisible by 3.

N=4: sum is 9 + 4 = 13, not divisible by 3.

N=6: sum is 15, divisible by 3.

N=8: sum is 9 + 8 = 17, not divisible by 3.

Therefore, only N=6 works.

Therefore, the answer is 6.

But let me check once more. Maybe I made a mistake in calculating the sum of digits? Let me add them again.

Digits: 2, 1, 4, 2, 0, N.

2 + 1 = 3

3 + 4 = 7

7 + 2 = 9

9 + 0 = 9

9 + N = 9 + N.

Yes, that's correct.

So 9 + N must be divisible by 3.

Since 9 is already divisible by 3, N must be divisible by 3.

So N is 3, 6, or 9. But N must be even. So only 6.

Therefore, N=6.

Therefore, the answer is 6.

I think that's solid.

**Final Answer**
The nonzero value of $ N $ is \boxed{6}.

To determine the nonzero value of $ N $ that makes the number $ 21420N $ divisible by 6, we must apply the rules of divisibility by 6. A number is divisible by 6 if and only if it is divisible by both 2 and 3.

---

### Step 1: Divisibility by 2

A number is divisible by 2 if its **last digit is even**. In this case, the number is $ 21420N $, and the last digit is $ N $. Therefore, for the number to be divisible by 2, $ N $ must be an **even digit**.

The possible even digits are:
$$
N \in \{2, 4, 6, 8\}
$$

---

### Step 2: Divisibility by 3

A number is divisible by 3 if the **sum of its digits is divisible by 3**. The digits of the number $ 21420N $ are:
$$
2, 1, 4, 2, 0, N
$$

Adding these digits:
$$
2 + 1 + 4 + 2 + 0 + N = 9 + N
$$

So, the sum of the digits is $ 9 + N $. For the number to be divisible by 3, $ 9 + N $ must be divisible by 3.

Since 9 is already divisible by 3, $ N $ must also be divisible by 3. That is:
$$
N \equiv 0 \pmod{3}
$$

---

### Step 3: Combine Both Conditions

From Step 1, $ N $ must be even.
From Step 2, $ N $ must be divisible by 3.

So, we are looking for a **nonzero digit** that is both **even** and **divisible by 3**.

The digits that satisfy both conditions are:
$$
N \in \{6\}
$$

(Other even digits: 2, 4, 8 — none of these are divisible by 3.)

---

### Step 4: Final Verification

Let’s verify that $ N = 6 $ works:

- The number becomes $ 214206 $
- Last digit is 6 → divisible by 2 ✅
- Sum of digits: $ 2 + 1 + 4 + 2 + 0 + 6 = 15 $, which is divisible by 3 ✅

Thus, the number is divisible by 6.

---

### Final Answer

$$
\boxed{6}
$$

Solution: The number $21420N$ is divisible by $6$ if and only if it is even and the sum of its digits is divisible by $3$. So $N$ must be even, and $2 + 1 + 4 + 2 + 0 + N = 9 + N$ must be divisible by $3$.  Since $9$ is divisible by $3$, we see that $N$ must also be divisible by $3$. The only digit that works is $N = \boxed{6}$.


Explanation:
Okay, so I need to simplify this expression: (10r³)(4r⁶) divided by 8r⁴. Let me think about how to approach this.

First, I remember that when you multiply terms with the same base, you add the exponents. But here, there's also division involved. Maybe I should handle the coefficients first and then the variables? Let me try that.

Starting with the coefficients: 10 multiplied by 4, and then divided by 8. Let me calculate that. 10 times 4 is 40. Then, 40 divided by 8 is... 5. Okay, so the coefficient part simplifies to 5. That seems straightforward.

Now, the variables. The numerator has r³ multiplied by r⁶. When you multiply exponents with the same base, you add the exponents. So, r³ times r⁶ would be r^(3+6) which is r⁹. Then, the denominator is r⁴. So, dividing r⁹ by r⁴. When you divide exponents with the same base, you subtract the exponents. So, r^(9-4) is r⁵.

Putting it all together, the coefficient is 5 and the variable part is r⁵. So, the simplified expression should be 5r⁵. Let me check if I did everything right.

Wait, let me verify each step again. First, coefficients: 10*4=40, then 40/8=5. Yep, that's correct. Then, for the variables: multiplying r³ and r⁶ gives r^(3+6)=r⁹. Then dividing by r⁴ gives r^(9-4)=r⁵. So, combining them gives 5r⁵.

Hmm, seems right. Let me think if there's another way to approach this. Maybe simplifying before multiplying? Let me try that.

Original expression: (10r³)(4r⁶)/(8r⁴). Maybe I can simplify the coefficients first. 10 divided by 8 is 5/4, and then multiplied by 4? Wait, let me see.

Alternatively, maybe factor out common terms. Let's see. 10 and 8 have a common factor of 2. So, 10 divided by 2 is 5, and 8 divided by 2 is 4. Then, 4 divided by 4 is 1. Wait, maybe that's complicating. Let me try again.

Breaking down the coefficients: 10/8 * 4. Which is (10/8)*4. Simplify 10/8 first: that's 5/4. Then 5/4 * 4 is 5. Oh, same result. So that works.

For the variables, same as before. So, yeah, same answer.

Alternatively, maybe write all the terms out. Let me write the numerator as 10*4*r³*r⁶ and denominator as 8*r⁴. Then, rearrange the terms: (10*4/8) * (r³*r⁶)/r⁴. Which is the same as before.

So, 10*4 is 40, divided by 8 is 5. Then, r³*r⁶ is r⁹, divided by r⁴ is r⁵. So, 5r⁵.

I think that's correct. Let me check with actual numbers. Let me pick a value for r and see if the original expression and the simplified one give the same result. Let's choose r=2.

Original expression: (10*(2)^3)*(4*(2)^6) divided by (8*(2)^4). Let me compute each part.

First, compute numerator: 10*(8) = 80, and 4*(64) = 256. Multiply those together: 80*256. Let me calculate that. 80*256. Well, 80*200=16,000 and 80*56=4,480. So total is 16,000 + 4,480 = 20,480.

Denominator: 8*(2)^4 = 8*16 = 128.

So, the entire expression is 20,480 divided by 128. Let me compute that. 128*160 = 20,480. So, 20,480 / 128 = 160.

Now, simplified expression: 5*(2)^5. 2^5 is 32. 5*32=160. Same result. So that checks out.

Another test with r=1. Original expression: (10*1^3)(4*1^6)/ (8*1^4) = (10*1)(4*1)/ (8*1) = 10*4 /8 = 40/8=5. Simplified expression: 5*(1)^5=5*1=5. Same result.

Another test with r=3. Original expression: numerator: 10*(27)*4*(729) = 10*4*27*729. Let me compute that. 10*4=40, 27*729. Hmm, 27*700=18,900 and 27*29=783, so total is 18,900 + 783 = 19,683. Then 40*19,683 = 787,320. Denominator: 8*(81)=648. So 787,320 / 648. Let me divide numerator and denominator by 8: 787,320 /8 = 98,415; 648 /8=81. So 98,415 /81. Let me divide 98,415 by 81. 81*1200=97,200. Subtract: 98,415 -97,200=1,215. 81*15=1,215. So total is 1200+15=1215.

Simplified expression: 5*(3)^5. 3^5=243. 5*243=1,215. Same result.

Okay, so that seems to check out. Therefore, my simplified expression is correct.

So, the answer is 5r⁵.

**Final Answer**
The simplified expression is \boxed{5r^5}.

To simplify the expression:

$$
\frac{(10r^3)(4r^6)}{8r^4}
$$

we will break the problem into two parts: simplifying the **coefficients** and simplifying the **variables**.

---

### Step 1: Simplify the Coefficients

We first handle the numerical coefficients:

$$
\frac{10 \cdot 4}{8} = \frac{40}{8} = 5
$$

So, the coefficient simplifies to **5**.

---

### Step 2: Simplify the Variables

Now, we simplify the powers of $ r $:

- In the numerator: $ r^3 \cdot r^6 = r^{3+6} = r^9 $
- In the denominator: $ r^4 $

Now divide the powers of $ r $:

$$
\frac{r^9}{r^4} = r^{9-4} = r^5
$$

---

### Step 3: Combine the Results

Now, combine the simplified coefficient and variable:

$$
5 \cdot r^5 = 5r^5
$$

---

### Final Answer

$$
\boxed{5r^5}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: (10r³)(4r⁶) divided by 8r⁴. Let me think about how to approach this.

First, I remember that when you multiply terms with the same base, you add the exponents. But here, there's also division involved. Maybe I should handle the coefficients first and then the variables? Let me try that.

Starting with the coefficients: 10 multiplied by 4, and then divided by 8. Let me calculate that. 10 times 4 is 40. Then, 40 divided by 8 is... 5. Okay, so the coefficient part simplifies to 5. That seems straightforward.

Now, the variables. The numerator has r³ multiplied by r⁶. When you multiply exponents with the same base, you add the exponents. So, r³ times r⁶ would be r^(3+6) which is r⁹. Then, the denominator is r⁴. So, dividing r⁹ by r⁴. When you divide exponents with the same base, you subtract the exponents. So, r^(9-4) is r⁵.

Putting it all together, the coefficient is 5 and the variable part is r⁵. So, the simplified expression should be 5r⁵. Let me check if I did everything right.

Wait, let me verify each step again. First, coefficients: 10*4=40, then 40/8=5. Yep, that's correct. Then, for the variables: multiplying r³ and r⁶ gives r^(3+6)=r⁹. Then dividing by r⁴ gives r^(9-4)=r⁵. So, combining them gives 5r⁵.

Hmm, seems right. Let me think if there's another way to approach this. Maybe simplifying before multiplying? Let me try that.

Original expression: (10r³)(4r⁶)/(8r⁴). Maybe I can simplify the coefficients first. 10 divided by 8 is 5/4, and then multiplied by 4? Wait, let me see.

Alternatively, maybe factor out common terms. Let's see. 10 and 8 have a common factor of 2. So, 10 divided by 2 is 5, and 8 divided by 2 is 4. Then, 4 divided by 4 is 1. Wait, maybe that's complicating. Let me try again.

Breaking down the coefficients: 10/8 * 4. Which is (10/8)*4. Simplify 10/8 first: that's 5/4. Then 5/4 * 4 is 5. Oh, same result. So that works.

For the variables, same as before. So, yeah, same answer.

Alternatively, maybe write all the terms out. Let me write the numerator as 10*4*r³*r⁶ and denominator as 8*r⁴. Then, rearrange the terms: (10*4/8) * (r³*r⁶)/r⁴. Which is the same as before.

So, 10*4 is 40, divided by 8 is 5. Then, r³*r⁶ is r⁹, divided by r⁴ is r⁵. So, 5r⁵.

I think that's correct. Let me check with actual numbers. Let me pick a value for r and see if the original expression and the simplified one give the same result. Let's choose r=2.

Original expression: (10*(2)^3)*(4*(2)^6) divided by (8*(2)^4). Let me compute each part.

First, compute numerator: 10*(8) = 80, and 4*(64) = 256. Multiply those together: 80*256. Let me calculate that. 80*256. Well, 80*200=16,000 and 80*56=4,480. So total is 16,000 + 4,480 = 20,480.

Denominator: 8*(2)^4 = 8*16 = 128.

So, the entire expression is 20,480 divided by 128. Let me compute that. 128*160 = 20,480. So, 20,480 / 128 = 160.

Now, simplified expression: 5*(2)^5. 2^5 is 32. 5*32=160. Same result. So that checks out.

Another test with r=1. Original expression: (10*1^3)(4*1^6)/ (8*1^4) = (10*1)(4*1)/ (8*1) = 10*4 /8 = 40/8=5. Simplified expression: 5*(1)^5=5*1=5. Same result.

Another test with r=3. Original expression: numerator: 10*(27)*4*(729) = 10*4*27*729. Let me compute that. 10*4=40, 27*729. Hmm, 27*700=18,900 and 27*29=783, so total is 18,900 + 783 = 19,683. Then 40*19,683 = 787,320. Denominator: 8*(81)=648. So 787,320 / 648. Let me divide numerator and denominator by 8: 787,320 /8 = 98,415; 648 /8=81. So 98,415 /81. Let me divide 98,415 by 81. 81*1200=97,200. Subtract: 98,415 -97,200=1,215. 81*15=1,215. So total is 1200+15=1215.

Simplified expression: 5*(3)^5. 3^5=243. 5*243=1,215. Same result.

Okay, so that seems to check out. Therefore, my simplified expression is correct.

So, the answer is 5r⁵.

**Final Answer**
The simplified expression is \boxed{5r^5}.

To simplify the expression:

$$
\frac{(10r^3)(4r^6)}{8r^4}
$$

we will break the problem into two parts: simplifying the **coefficients** and simplifying the **variables**.

---

### Step 1: Simplify the Coefficients

We first handle the numerical coefficients:

$$
\frac{10 \cdot 4}{8} = \frac{40}{8} = 5
$$

So, the coefficient simplifies to **5**.

---

### Step 2: Simplify the Variables

Now, we simplify the powers of $ r $:

- In the numerator: $ r^3 \cdot r^6 = r^{3+6} = r^9 $
- In the denominator: $ r^4 $

Now divide the powers of $ r $:

$$
\frac{r^9}{r^4} = r^{9-4} = r^5
$$

---

### Step 3: Combine the Results

Now, combine the simplified coefficient and variable:

$$
5 \cdot r^5 = 5r^5
$$

---

### Final Answer

$$
\boxed{5r^5}
$$

Solution: We have  \[\frac{(10r^3)(4r^6)}{8r^4}= \frac{40r^{3+6}}{8r^4} = \frac{40}{8}r^{3+6-4} = \boxed{5r^5}.\]


Explanation:
Okay, so I need to figure out the cost of a pencil in cents. The problem says that three pencils and a jumbo eraser cost $1.24, and five pencils and a jumbo eraser cost $1.82. Hmm, let me think about how to approach this.

First, I remember that when you have two equations with two variables, you can solve them using substitution or elimination. Since both equations have a jumbo eraser, maybe I can subtract one equation from the other to eliminate the eraser cost. Let me write down the equations to make it clearer.

Let me denote the cost of a pencil as "p" cents and the cost of a jumbo eraser as "e" cents. Wait, but the problem mentions dollars, so maybe I should convert everything to cents to avoid dealing with decimals? That might make calculations easier. Let me check the original prices:

Three pencils and a jumbo eraser cost $1.24. Converting that to cents, it's 124 cents. Similarly, five pencils and a jumbo eraser cost $1.82, which is 182 cents.

So, the equations would be:

1) 3p + e = 124

2) 5p + e = 182

Okay, so now I have two equations with two variables. If I subtract the first equation from the second, the e terms should cancel out. Let me try that.

Subtracting equation 1 from equation 2:

(5p + e) - (3p + e) = 182 - 124

Let me compute the left side:

5p + e - 3p - e = (5p - 3p) + (e - e) = 2p + 0 = 2p

On the right side, 182 - 124. Let me calculate that. 182 minus 124... 182 - 100 is 82, then minus 24 more is 58. So 58.

Therefore, 2p = 58. Then, solving for p, divide both sides by 2: p = 58 / 2 = 29.

Wait, so the cost of a pencil is 29 cents? Let me check if that makes sense. Let me plug this back into one of the original equations to find the cost of the eraser and verify.

Let me use equation 1: 3p + e = 124. If p is 29, then 3*29 is 87. So 87 + e = 124. Therefore, e = 124 - 87 = 37. So the eraser would cost 37 cents.

Let me check with equation 2: 5p + e = 5*29 + 37. 5*29 is 145, plus 37 is 182. Which matches the second equation. Perfect! So that seems to check out.

But wait, the problem says "In cents, what is the cost of a pencil?" So since we found p = 29 cents, that should be the answer. But let me just make sure I didn't make any calculation errors.

Let me recheck the subtraction step. Original equations:

Equation 1: 3p + e = 124

Equation 2: 5p + e = 182

Subtract equation 1 from equation 2:

(5p + e) - (3p + e) = 182 - 124

Which simplifies to 2p = 58, so p = 29. Yes, that's correct.

Alternatively, maybe I can solve using substitution. Let me try that method to confirm.

From equation 1: e = 124 - 3p

Then substitute into equation 2:

5p + (124 - 3p) = 182

Simplify: 5p + 124 - 3p = 182 => 2p + 124 = 182 => 2p = 182 - 124 = 58 => p = 29. Same result.

So that's consistent. Therefore, the cost of a pencil is 29 cents.

But just to be thorough, let me check if there's any possibility that the problem expects something else. For example, maybe the prices are in dollars and cents, but the question says "No prices include tax," so we don't have to worry about tax. Also, the problem says "In cents," so the answer should be in cents, which we have as 29.

Wait, but let me check if converting dollars to cents was done correctly. $1.24 is 124 cents, yes. $1.82 is 182 cents, yes. So that's correct.

Alternatively, if I had kept the prices in dollars, would that have made a difference? Let me try that approach as well.

Let me denote p as dollars and e as dollars. Then:

Equation 1: 3p + e = 1.24

Equation 2: 5p + e = 1.82

Subtract equation 1 from equation 2:

(5p + e) - (3p + e) = 1.82 - 1.24 => 2p = 0.58 => p = 0.29 dollars. Which is 29 cents. So same answer.

Therefore, regardless of whether I convert to cents first or work in dollars, I get the same result.

So, I think that's solid. The answer is 29 cents.

But just to make sure, let me think if there's any other way this could be interpreted. For example, maybe the problem is trying to trick me with something? Like, maybe the eraser is jumbo, so maybe it's a different price? But no, the problem says "jumbo eraser" but doesn't mention any different pricing for jumbo, so I think it's just a name, not affecting the price.

Alternatively, maybe there's a typo in the problem? But assuming the problem is stated correctly, then my answer should be correct.

Another check: if a pencil is 29 cents, then three pencils are 87 cents, plus eraser 37 cents, total 124 cents, which is $1.24. Five pencils would be 145 cents, plus eraser 37 cents, total 182 cents, which is $1.82. Perfect.

So, I think that's correct. Therefore, the cost of a pencil is 29 cents.

**Final Answer**
The cost of a pencil is \boxed{29} cents.

To determine the cost of a pencil, we are given two equations based on the total cost of pencils and a jumbo eraser:

- Three pencils and a jumbo eraser cost \$1.24.
- Five pencils and a jumbo eraser cost \$1.82.

---

### Step 1: Convert Dollars to Cents

To simplify calculations, we convert the dollar amounts to cents:

- \$1.24 = 124 cents
- \$1.82 = 182 cents

Let:
- $ p $ = cost of one pencil (in cents)
- $ e $ = cost of one jumbo eraser (in cents)

We can now write the system of equations:

$$
\begin{align*}
(1) & \quad 3p + e = 124 \\
(2) & \quad 5p + e = 182
\end{align*}
$$

---

### Step 2: Eliminate the Eraser Cost

Subtract equation (1) from equation (2):

$$
(5p + e) - (3p + e) = 182 - 124
$$

Simplify both sides:

$$
2p = 58
$$

Solve for $ p $:

$$
p = \frac{58}{2} = 29
$$

---

### Step 3: Verify the Solution

Plug $ p = 29 $ back into equation (1) to find $ e $:

$$
3(29) + e = 124 \Rightarrow 87 + e = 124 \Rightarrow e = 37
$$

Now verify with equation (2):

$$
5(29) + 37 = 145 + 37 = 182
$$

Both equations are satisfied, confirming the solution is correct.

---

### Final Answer

The cost of a pencil is:

$$
\boxed{29}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the cost of a pencil in cents. The problem says that three pencils and a jumbo eraser cost $1.24, and five pencils and a jumbo eraser cost $1.82. Hmm, let me think about how to approach this.

First, I remember that when you have two equations with two variables, you can solve them using substitution or elimination. Since both equations have a jumbo eraser, maybe I can subtract one equation from the other to eliminate the eraser cost. Let me write down the equations to make it clearer.

Let me denote the cost of a pencil as "p" cents and the cost of a jumbo eraser as "e" cents. Wait, but the problem mentions dollars, so maybe I should convert everything to cents to avoid dealing with decimals? That might make calculations easier. Let me check the original prices:

Three pencils and a jumbo eraser cost $1.24. Converting that to cents, it's 124 cents. Similarly, five pencils and a jumbo eraser cost $1.82, which is 182 cents.

So, the equations would be:

1) 3p + e = 124

2) 5p + e = 182

Okay, so now I have two equations with two variables. If I subtract the first equation from the second, the e terms should cancel out. Let me try that.

Subtracting equation 1 from equation 2:

(5p + e) - (3p + e) = 182 - 124

Let me compute the left side:

5p + e - 3p - e = (5p - 3p) + (e - e) = 2p + 0 = 2p

On the right side, 182 - 124. Let me calculate that. 182 minus 124... 182 - 100 is 82, then minus 24 more is 58. So 58.

Therefore, 2p = 58. Then, solving for p, divide both sides by 2: p = 58 / 2 = 29.

Wait, so the cost of a pencil is 29 cents? Let me check if that makes sense. Let me plug this back into one of the original equations to find the cost of the eraser and verify.

Let me use equation 1: 3p + e = 124. If p is 29, then 3*29 is 87. So 87 + e = 124. Therefore, e = 124 - 87 = 37. So the eraser would cost 37 cents.

Let me check with equation 2: 5p + e = 5*29 + 37. 5*29 is 145, plus 37 is 182. Which matches the second equation. Perfect! So that seems to check out.

But wait, the problem says "In cents, what is the cost of a pencil?" So since we found p = 29 cents, that should be the answer. But let me just make sure I didn't make any calculation errors.

Let me recheck the subtraction step. Original equations:

Equation 1: 3p + e = 124

Equation 2: 5p + e = 182

Subtract equation 1 from equation 2:

(5p + e) - (3p + e) = 182 - 124

Which simplifies to 2p = 58, so p = 29. Yes, that's correct.

Alternatively, maybe I can solve using substitution. Let me try that method to confirm.

From equation 1: e = 124 - 3p

Then substitute into equation 2:

5p + (124 - 3p) = 182

Simplify: 5p + 124 - 3p = 182 => 2p + 124 = 182 => 2p = 182 - 124 = 58 => p = 29. Same result.

So that's consistent. Therefore, the cost of a pencil is 29 cents.

But just to be thorough, let me check if there's any possibility that the problem expects something else. For example, maybe the prices are in dollars and cents, but the question says "No prices include tax," so we don't have to worry about tax. Also, the problem says "In cents," so the answer should be in cents, which we have as 29.

Wait, but let me check if converting dollars to cents was done correctly. $1.24 is 124 cents, yes. $1.82 is 182 cents, yes. So that's correct.

Alternatively, if I had kept the prices in dollars, would that have made a difference? Let me try that approach as well.

Let me denote p as dollars and e as dollars. Then:

Equation 1: 3p + e = 1.24

Equation 2: 5p + e = 1.82

Subtract equation 1 from equation 2:

(5p + e) - (3p + e) = 1.82 - 1.24 => 2p = 0.58 => p = 0.29 dollars. Which is 29 cents. So same answer.

Therefore, regardless of whether I convert to cents first or work in dollars, I get the same result.

So, I think that's solid. The answer is 29 cents.

But just to make sure, let me think if there's any other way this could be interpreted. For example, maybe the problem is trying to trick me with something? Like, maybe the eraser is jumbo, so maybe it's a different price? But no, the problem says "jumbo eraser" but doesn't mention any different pricing for jumbo, so I think it's just a name, not affecting the price.

Alternatively, maybe there's a typo in the problem? But assuming the problem is stated correctly, then my answer should be correct.

Another check: if a pencil is 29 cents, then three pencils are 87 cents, plus eraser 37 cents, total 124 cents, which is $1.24. Five pencils would be 145 cents, plus eraser 37 cents, total 182 cents, which is $1.82. Perfect.

So, I think that's correct. Therefore, the cost of a pencil is 29 cents.

**Final Answer**
The cost of a pencil is \boxed{29} cents.

To determine the cost of a pencil, we are given two equations based on the total cost of pencils and a jumbo eraser:

- Three pencils and a jumbo eraser cost \$1.24.
- Five pencils and a jumbo eraser cost \$1.82.

---

### Step 1: Convert Dollars to Cents

To simplify calculations, we convert the dollar amounts to cents:

- \$1.24 = 124 cents
- \$1.82 = 182 cents

Let:
- $ p $ = cost of one pencil (in cents)
- $ e $ = cost of one jumbo eraser (in cents)

We can now write the system of equations:

$$
\begin{align*}
(1) & \quad 3p + e = 124 \\
(2) & \quad 5p + e = 182
\end{align*}
$$

---

### Step 2: Eliminate the Eraser Cost

Subtract equation (1) from equation (2):

$$
(5p + e) - (3p + e) = 182 - 124
$$

Simplify both sides:

$$
2p = 58
$$

Solve for $ p $:

$$
p = \frac{58}{2} = 29
$$

---

### Step 3: Verify the Solution

Plug $ p = 29 $ back into equation (1) to find $ e $:

$$
3(29) + e = 124 \Rightarrow 87 + e = 124 \Rightarrow e = 37
$$

Now verify with equation (2):

$$
5(29) + 37 = 145 + 37 = 182
$$

Both equations are satisfied, confirming the solution is correct.

---

### Final Answer

The cost of a pencil is:

$$
\boxed{29}
$$

Solution:
Call the price of a pencil $p$ and the price of a jumbo eraser $e$, in terms of cents. We can use the following system of equations to represent the information given: \begin{align*}
3p + e &= 124 \\
5p + e &= 182 \\
\end{align*} Subtracting the first equation from the second gives $2p = 58$, or $p = 29$. Thus, the cost of a pencil is $\boxed{29}$ cents.


Explanation:
Okay, so I need to figure out what 9 divided by 2 is as a decimal. Let me think. Hmm, I remember that dividing by 2 is the same as multiplying by 0.5, right? So maybe I can just do 9 times 0.5? Let me check that.

Wait, 9 times 0.5 is the same as half of 9. Half of 9 is 4.5, so is that the answer? But let me make sure I'm not making a mistake here. Maybe I should do the division step by step.

Alright, let's try dividing 9 by 2. So 2 goes into 9 how many times? Well, 2 times 4 is 8, which is less than 9. Then 2 times 5 is 10, which is too much. So it goes in 4 times with a remainder. So 4 times 2 is 8, subtract that from 9, and I get a remainder of 1.

Now, since we're dealing with decimals, I can add a decimal point and some zeros to the 9. So 9 divided by 2 is 4. Then, bringing down a zero, it becomes 10. Now, 2 goes into 10 exactly 5 times. So that gives me 4.5.

Wait, so that's the same as before. So 9 divided by 2 is 4.5. Let me verify with another method. Maybe converting the fraction to a decimal by expanding it.

The fraction is 9/2. If I multiply numerator and denominator by 5, that would make the denominator 10, right? Because 2 times 5 is 10. So if I do that, the numerator becomes 9 times 5, which is 45. So 45/10 is 4.5. Yep, that checks out.

Another way to think about it is that 1/2 is 0.5, so 9 times 0.5 is 4.5. All these methods are giving me the same answer. Maybe I can also use a calculator to confirm? But since I don't have a calculator here, I need to rely on my math.

Let me try long division again just to be thorough. Dividing 9 by 2.

2 | 9.0

2 goes into 9 four times (4*2=8). Subtract 8 from 9, you get 1. Bring down a 0, making it 10. 2 goes into 10 five times (5*2=10). Subtract 10 from 10, you get 0. So the division ends here. So the result is 4.5.

No remainder, so it's a terminating decimal. That makes sense.

Alternatively, thinking in terms of fractions: 9/2 is equal to 4 and 1/2. Since 1/2 is 0.5, adding that to 4 gives 4.5.

All these different approaches are leading me to the same conclusion. I don't see any mistakes in my reasoning. So I think the decimal form of 9/2 is definitely 4.5.

Just to recap: 9 divided by 2 is 4.5. Converting the fraction by multiplying numerator and denominator by 5 gives 45/10, which is 4.5. Using decimal multiplication, 9 times 0.5 is 4.5. Long division confirms it. And converting the mixed number also gives the same result.

Therefore, I can confidently say that 9/2 as a decimal is 4.5.

**Final Answer**
The decimal form of $\frac{9}{2}$ is \boxed{4.5}.

To express the fraction $\frac{9}{2}$ as a decimal, we can approach the problem using several equivalent methods, all of which lead to the same result.

---

### **Method 1: Division**

We can perform the division directly:

$$
\frac{9}{2} = 9 \div 2
$$

- 2 goes into 9 **4 times** (since $2 \times 4 = 8$).
- Subtract 8 from 9, leaving a remainder of 1.
- Bring down a 0, making it 10.
- 2 goes into 10 **5 times** (since $2 \times 5 = 10$).
- Subtract 10 from 10, leaving no remainder.

So, the result is:

$$
\frac{9}{2} = 4.5
$$

---

### **Method 2: Decimal Conversion**

We can convert the fraction by multiplying both numerator and denominator by 5 to make the denominator 10:

$$
\frac{9}{2} = \frac{9 \times 5}{2 \times 5} = \frac{45}{10} = 4.5
$$

---

### **Method 3: Fraction to Decimal via Known Values**

We know that:

$$
\frac{1}{2} = 0.5
$$

So:

$$
\frac{9}{2} = 9 \times \frac{1}{2} = 9 \times 0.5 = 4.5
$$

---

### **Conclusion**

All methods consistently show that the decimal representation of $\frac{9}{2}$ is:

$$
\boxed{4.5}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what 9 divided by 2 is as a decimal. Let me think. Hmm, I remember that dividing by 2 is the same as multiplying by 0.5, right? So maybe I can just do 9 times 0.5? Let me check that.

Wait, 9 times 0.5 is the same as half of 9. Half of 9 is 4.5, so is that the answer? But let me make sure I'm not making a mistake here. Maybe I should do the division step by step.

Alright, let's try dividing 9 by 2. So 2 goes into 9 how many times? Well, 2 times 4 is 8, which is less than 9. Then 2 times 5 is 10, which is too much. So it goes in 4 times with a remainder. So 4 times 2 is 8, subtract that from 9, and I get a remainder of 1.

Now, since we're dealing with decimals, I can add a decimal point and some zeros to the 9. So 9 divided by 2 is 4. Then, bringing down a zero, it becomes 10. Now, 2 goes into 10 exactly 5 times. So that gives me 4.5.

Wait, so that's the same as before. So 9 divided by 2 is 4.5. Let me verify with another method. Maybe converting the fraction to a decimal by expanding it.

The fraction is 9/2. If I multiply numerator and denominator by 5, that would make the denominator 10, right? Because 2 times 5 is 10. So if I do that, the numerator becomes 9 times 5, which is 45. So 45/10 is 4.5. Yep, that checks out.

Another way to think about it is that 1/2 is 0.5, so 9 times 0.5 is 4.5. All these methods are giving me the same answer. Maybe I can also use a calculator to confirm? But since I don't have a calculator here, I need to rely on my math.

Let me try long division again just to be thorough. Dividing 9 by 2.

2 | 9.0

2 goes into 9 four times (4*2=8). Subtract 8 from 9, you get 1. Bring down a 0, making it 10. 2 goes into 10 five times (5*2=10). Subtract 10 from 10, you get 0. So the division ends here. So the result is 4.5.

No remainder, so it's a terminating decimal. That makes sense.

Alternatively, thinking in terms of fractions: 9/2 is equal to 4 and 1/2. Since 1/2 is 0.5, adding that to 4 gives 4.5.

All these different approaches are leading me to the same conclusion. I don't see any mistakes in my reasoning. So I think the decimal form of 9/2 is definitely 4.5.

Just to recap: 9 divided by 2 is 4.5. Converting the fraction by multiplying numerator and denominator by 5 gives 45/10, which is 4.5. Using decimal multiplication, 9 times 0.5 is 4.5. Long division confirms it. And converting the mixed number also gives the same result.

Therefore, I can confidently say that 9/2 as a decimal is 4.5.

**Final Answer**
The decimal form of $\frac{9}{2}$ is \boxed{4.5}.

To express the fraction $\frac{9}{2}$ as a decimal, we can approach the problem using several equivalent methods, all of which lead to the same result.

---

### **Method 1: Division**

We can perform the division directly:

$$
\frac{9}{2} = 9 \div 2
$$

- 2 goes into 9 **4 times** (since $2 \times 4 = 8$).
- Subtract 8 from 9, leaving a remainder of 1.
- Bring down a 0, making it 10.
- 2 goes into 10 **5 times** (since $2 \times 5 = 10$).
- Subtract 10 from 10, leaving no remainder.

So, the result is:

$$
\frac{9}{2} = 4.5
$$

---

### **Method 2: Decimal Conversion**

We can convert the fraction by multiplying both numerator and denominator by 5 to make the denominator 10:

$$
\frac{9}{2} = \frac{9 \times 5}{2 \times 5} = \frac{45}{10} = 4.5
$$

---

### **Method 3: Fraction to Decimal via Known Values**

We know that:

$$
\frac{1}{2} = 0.5
$$

So:

$$
\frac{9}{2} = 9 \times \frac{1}{2} = 9 \times 0.5 = 4.5
$$

---

### **Conclusion**

All methods consistently show that the decimal representation of $\frac{9}{2}$ is:

$$
\boxed{4.5}
$$

Solution: We can solve this problem by division.  Alternatively, we can multiply the numerator and denominator by 5, yielding $\frac{45}{10}$.  Since dividing a number by 10 shifts the decimal point to the left by one place, this yields $\boxed{4.5}$.


Explanation:
Okay, so I need to convert the fraction 57/160 into a terminating decimal. Hmm, let me think about how to do this. First, I remember that a fraction will terminate if the denominator, after simplifying the fraction, has only 2 and/or 5 as prime factors. Let me check if 160 meets that condition.

Breaking down 160 into prime factors: 160 divided by 2 is 80, divided by 2 again is 40, again is 20, again is 10, again is 5. So that's 2^5 * 5^1. Yep, only 2s and 5s. So that means 57/160 should be a terminating decimal. Good.

Now, how do I convert it? Well, one way is to divide 57 by 160 directly. But maybe there's a smarter way? Let me think. Since the denominator is 160, which is 16 * 10, or 2^5 * 5^1. To make it a power of 10, I can multiply numerator and denominator by something to get the denominator to be 10^n.

Let me try that. The denominator is 160. Let me see, 160 is 16 * 10, so if I multiply numerator and denominator by 5, then denominator becomes 160 * 5 = 800, which is 8 * 100, but that might not be helpful. Wait, maybe I need to get the denominator to be 10^something. Let me think.

Since 160 = 2^5 * 5^1, to make it a power of 10, I need to have the same number of 2s and 5s. So, since there are 5 2s and 1 5, I need to multiply by 5^4 to make the denominator 2^5 * 5^5 = 10^5. So, multiplying numerator and denominator by 5^4, which is 625.

So, 57 * 625 divided by 160 * 625. Let me compute that.

First, 57 * 625. Let me calculate that. 57 * 625. Hmm, 625 is 5^4, which is 5*5*5*5. Alternatively, 625 is 500 + 125. So 57*625 = 57*(500 + 125) = 57*500 + 57*125.

Calculating 57*500: 57*5*100 = 285*100 = 28,500.

Then 57*125: 57*(100 + 25) = 57*100 + 57*25 = 5,700 + 1,425 = 7,125.

Adding those together: 28,500 + 7,125 = 35,625.

So the numerator is 35,625. The denominator is 160*625. Let me compute that. 160*625. Well, 160*625. Since 625 is 5^4, and 160 is 16*10. So 16*10*625 = 16*625*10. 16*625: 16*600 = 9,600 and 16*25 = 400, so total is 9,600 + 400 = 10,000. Then times 10 is 100,000.

So denominator is 100,000. Therefore, the fraction is 35,625 / 100,000. Which is 0.35625.

Wait, let me check that. 35,625 divided by 100,000. Moving the decimal five places to the left: 35,625 becomes 0.35625. Yes. So that's the decimal.

Alternatively, maybe I can do this division step by step. Let me try dividing 57 by 160.

Since 57 is less than 160, we write it as 57.000000... divided by 160.

So, 160 goes into 570 how many times? Let me calculate. 160*3=480, 160*4=640. 640 is too big, so 3 times. 3*160=480. Subtract 480 from 570: 570-480=90. Bring down a zero: 900.

Now, 160 goes into 900 how many times? 160*5=800, 160*6=960. 960 is too big, so 5 times. 5*160=800. Subtract from 900: 100. Bring down a zero: 1000.

160 goes into 1000 how many times? 160*6=960, 160*6=960. Subtract 960 from 1000: 40. Bring down a zero: 400.

160 goes into 400 exactly 2 times. 2*160=320. Subtract from 400: 80. Bring down a zero: 800.

160 goes into 800 exactly 5 times. 5*160=800. Subtract, remainder 0.

So putting that all together:

First step: 3, then 5, then 6, then 2, then 5. So the decimal is 0.35625.

So that matches the previous result.

Alternatively, maybe I can use another method. Since 1/160 is equal to... Well, 1/160 is 0.00625. Wait, let me check that. 1/160. Since 1/10 is 0.1, 1/16 is 0.0625. So 1/160 is 1/(16*10) = (1/16)/10 = 0.0625/10 = 0.00625.

So if 1/160 is 0.00625, then 57/160 is 57 * 0.00625. Let me compute that.

Calculating 57 * 0.00625. Let me think. 0.00625 is 6.25 * 10^-3. So 57 * 6.25 * 10^-3.

First compute 57 * 6.25.

Well, 57 * 6 = 342, 57 * 0.25 = 14.25. So total is 342 + 14.25 = 356.25. Then multiply by 10^-3: 0.35625.

Same result.

So all methods lead to 0.35625.

Therefore, the decimal is terminating, as expected, and the value is 0.35625.

Let me just check once more with another approach. Let me convert the fraction step by step.

Multiply numerator and denominator by 5 to make the denominator 800.

Wait, 160 * 5 = 800. So 57 * 5 = 285. So 285/800.

Then, 800 is 8 * 100. So divide numerator and denominator by 8? Wait, 285 divided by 8? Maybe not helpful. Alternatively, 800 is 10^3 * 0.8? Not sure.

Alternatively, divide 285 by 800. Let me do that division.

285 divided by 800. Since 285 is less than 800, it's 0.???

Multiply numerator and denominator by 10: 2850 / 8000.

Divide numerator and denominator by 10: 285 / 800.

Wait, same as before. Maybe another way.

Alternatively, divide numerator and denominator by 5: 285 / 800 = 57 / 160. Back to original.

Alternatively, maybe convert 285/800. Let me divide 285 by 800.

Well, 800 goes into 285 zero times. Add decimal: 0.

800 goes into 2850 three times (3*800=2400). Subtract 2400 from 2850: 450. Bring down a zero: 4500.

800 goes into 4500 five times (5*800=4000). Subtract: 4500-4000=500. Bring down a zero: 5000.

800 goes into 5000 six times (6*800=4800). Subtract: 5000-4800=200. Bring down a zero: 2000.

800 goes into 2000 two times (2*800=1600). Subtract: 2000-1600=400. Bring down a zero: 4000.

800 goes into 4000 five times (5*800=4000). Subtract: 0.

So the decimal is 0.35625.

Same result again.

So regardless of the method, I get 0.35625.

Therefore, I can be confident that 57/160 as a decimal is 0.35625, which is a terminating decimal.

Just to recap, the key steps were recognizing that the denominator factors into 2s and 5s, which means the decimal will terminate. Then, using different methods—multiplying numerator and denominator to get a denominator of 10^n, performing long division, or breaking down the fraction into known decimal equivalents—all led to the same result.

I think that's thorough enough. I don't see any mistakes in my reasoning, and all methods concur. So the answer should be 0.35625.

**Final Answer**
The decimal representation of $\frac{57}{160}$ is \boxed{0.35625}.

To convert the fraction $\frac{57}{160}$ into a **terminating decimal**, we begin by analyzing the denominator.

---

### Step 1: Check for Terminating Decimal

A fraction will have a **terminating decimal** if, after simplifying, the **denominator only contains the prime factors 2 and/or 5**.

Let’s factor the denominator:

$$
160 = 2^5 \times 5
$$

Since the prime factors are only 2 and 5, the fraction **$\frac{57}{160}$** will indeed result in a **terminating decimal**.

---

### Step 2: Convert to a Decimal

There are multiple methods to convert the fraction to a decimal. One efficient way is to **multiply numerator and denominator by a power of 10** to make the denominator a power of 10.

We want to make the denominator $10^n$. Since $160 = 2^5 \times 5$, we need to multiply by $5^4 = 625$ to get:

$$
160 \times 625 = 100,000
$$

Now, multiply the numerator by the same factor:

$$
57 \times 625 = 35,625
$$

So the fraction becomes:

$$
\frac{35,625}{100,000}
$$

This is equivalent to:

$$
0.35625
$$

---

### Step 3: Verification via Long Division

To confirm, we can perform the division $57 \div 160$ directly:

- $160$ goes into $570$ three times (3 × 160 = 480), remainder 90.
- Bring down a zero: 900. $160$ goes into 900 five times (5 × 160 = 800), remainder 100.
- Bring down a zero: 1000. $160$ goes into 1000 six times (6 × 160 = 960), remainder 40.
- Bring down a zero: 400. $160$ goes into 400 two times (2 × 160 = 320), remainder 80.
- Bring down a zero: 800. $160$ goes into 800 five times (5 × 160 = 800), remainder 0.

So the decimal is:

$$
0.35625
$$

---

### Final Answer

$$
\boxed{0.35625}
$$

[Assistant]
Explanation:
Okay, so I need to convert the fraction 57/160 into a terminating decimal. Hmm, let me think about how to do this. First, I remember that a fraction will terminate if the denominator, after simplifying the fraction, has only 2 and/or 5 as prime factors. Let me check if 160 meets that condition.

Breaking down 160 into prime factors: 160 divided by 2 is 80, divided by 2 again is 40, again is 20, again is 10, again is 5. So that's 2^5 * 5^1. Yep, only 2s and 5s. So that means 57/160 should be a terminating decimal. Good.

Now, how do I convert it? Well, one way is to divide 57 by 160 directly. But maybe there's a smarter way? Let me think. Since the denominator is 160, which is 16 * 10, or 2^5 * 5^1. To make it a power of 10, I can multiply numerator and denominator by something to get the denominator to be 10^n.

Let me try that. The denominator is 160. Let me see, 160 is 16 * 10, so if I multiply numerator and denominator by 5, then denominator becomes 160 * 5 = 800, which is 8 * 100, but that might not be helpful. Wait, maybe I need to get the denominator to be 10^something. Let me think.

Since 160 = 2^5 * 5^1, to make it a power of 10, I need to have the same number of 2s and 5s. So, since there are 5 2s and 1 5, I need to multiply by 5^4 to make the denominator 2^5 * 5^5 = 10^5. So, multiplying numerator and denominator by 5^4, which is 625.

So, 57 * 625 divided by 160 * 625. Let me compute that.

First, 57 * 625. Let me calculate that. 57 * 625. Hmm, 625 is 5^4, which is 5*5*5*5. Alternatively, 625 is 500 + 125. So 57*625 = 57*(500 + 125) = 57*500 + 57*125.

Calculating 57*500: 57*5*100 = 285*100 = 28,500.

Then 57*125: 57*(100 + 25) = 57*100 + 57*25 = 5,700 + 1,425 = 7,125.

Adding those together: 28,500 + 7,125 = 35,625.

So the numerator is 35,625. The denominator is 160*625. Let me compute that. 160*625. Well, 160*625. Since 625 is 5^4, and 160 is 16*10. So 16*10*625 = 16*625*10. 16*625: 16*600 = 9,600 and 16*25 = 400, so total is 9,600 + 400 = 10,000. Then times 10 is 100,000.

So denominator is 100,000. Therefore, the fraction is 35,625 / 100,000. Which is 0.35625.

Wait, let me check that. 35,625 divided by 100,000. Moving the decimal five places to the left: 35,625 becomes 0.35625. Yes. So that's the decimal.

Alternatively, maybe I can do this division step by step. Let me try dividing 57 by 160.

Since 57 is less than 160, we write it as 57.000000... divided by 160.

So, 160 goes into 570 how many times? Let me calculate. 160*3=480, 160*4=640. 640 is too big, so 3 times. 3*160=480. Subtract 480 from 570: 570-480=90. Bring down a zero: 900.

Now, 160 goes into 900 how many times? 160*5=800, 160*6=960. 960 is too big, so 5 times. 5*160=800. Subtract from 900: 100. Bring down a zero: 1000.

160 goes into 1000 how many times? 160*6=960, 160*6=960. Subtract 960 from 1000: 40. Bring down a zero: 400.

160 goes into 400 exactly 2 times. 2*160=320. Subtract from 400: 80. Bring down a zero: 800.

160 goes into 800 exactly 5 times. 5*160=800. Subtract, remainder 0.

So putting that all together:

First step: 3, then 5, then 6, then 2, then 5. So the decimal is 0.35625.

So that matches the previous result.

Alternatively, maybe I can use another method. Since 1/160 is equal to... Well, 1/160 is 0.00625. Wait, let me check that. 1/160. Since 1/10 is 0.1, 1/16 is 0.0625. So 1/160 is 1/(16*10) = (1/16)/10 = 0.0625/10 = 0.00625.

So if 1/160 is 0.00625, then 57/160 is 57 * 0.00625. Let me compute that.

Calculating 57 * 0.00625. Let me think. 0.00625 is 6.25 * 10^-3. So 57 * 6.25 * 10^-3.

First compute 57 * 6.25.

Well, 57 * 6 = 342, 57 * 0.25 = 14.25. So total is 342 + 14.25 = 356.25. Then multiply by 10^-3: 0.35625.

Same result.

So all methods lead to 0.35625.

Therefore, the decimal is terminating, as expected, and the value is 0.35625.

Let me just check once more with another approach. Let me convert the fraction step by step.

Multiply numerator and denominator by 5 to make the denominator 800.

Wait, 160 * 5 = 800. So 57 * 5 = 285. So 285/800.

Then, 800 is 8 * 100. So divide numerator and denominator by 8? Wait, 285 divided by 8? Maybe not helpful. Alternatively, 800 is 10^3 * 0.8? Not sure.

Alternatively, divide 285 by 800. Let me do that division.

285 divided by 800. Since 285 is less than 800, it's 0.???

Multiply numerator and denominator by 10: 2850 / 8000.

Divide numerator and denominator by 10: 285 / 800.

Wait, same as before. Maybe another way.

Alternatively, divide numerator and denominator by 5: 285 / 800 = 57 / 160. Back to original.

Alternatively, maybe convert 285/800. Let me divide 285 by 800.

Well, 800 goes into 285 zero times. Add decimal: 0.

800 goes into 2850 three times (3*800=2400). Subtract 2400 from 2850: 450. Bring down a zero: 4500.

800 goes into 4500 five times (5*800=4000). Subtract: 4500-4000=500. Bring down a zero: 5000.

800 goes into 5000 six times (6*800=4800). Subtract: 5000-4800=200. Bring down a zero: 2000.

800 goes into 2000 two times (2*800=1600). Subtract: 2000-1600=400. Bring down a zero: 4000.

800 goes into 4000 five times (5*800=4000). Subtract: 0.

So the decimal is 0.35625.

Same result again.

So regardless of the method, I get 0.35625.

Therefore, I can be confident that 57/160 as a decimal is 0.35625, which is a terminating decimal.

Just to recap, the key steps were recognizing that the denominator factors into 2s and 5s, which means the decimal will terminate. Then, using different methods—multiplying numerator and denominator to get a denominator of 10^n, performing long division, or breaking down the fraction into known decimal equivalents—all led to the same result.

I think that's thorough enough. I don't see any mistakes in my reasoning, and all methods concur. So the answer should be 0.35625.

**Final Answer**
The decimal representation of $\frac{57}{160}$ is \boxed{0.35625}.

To convert the fraction $\frac{57}{160}$ into a **terminating decimal**, we begin by analyzing the denominator.

---

### Step 1: Check for Terminating Decimal

A fraction will have a **terminating decimal** if, after simplifying, the **denominator only contains the prime factors 2 and/or 5**.

Let’s factor the denominator:

$$
160 = 2^5 \times 5
$$

Since the prime factors are only 2 and 5, the fraction **$\frac{57}{160}$** will indeed result in a **terminating decimal**.

---

### Step 2: Convert to a Decimal

There are multiple methods to convert the fraction to a decimal. One efficient way is to **multiply numerator and denominator by a power of 10** to make the denominator a power of 10.

We want to make the denominator $10^n$. Since $160 = 2^5 \times 5$, we need to multiply by $5^4 = 625$ to get:

$$
160 \times 625 = 100,000
$$

Now, multiply the numerator by the same factor:

$$
57 \times 625 = 35,625
$$

So the fraction becomes:

$$
\frac{35,625}{100,000}
$$

This is equivalent to:

$$
0.35625
$$

---

### Step 3: Verification via Long Division

To confirm, we can perform the division $57 \div 160$ directly:

- $160$ goes into $570$ three times (3 × 160 = 480), remainder 90.
- Bring down a zero: 900. $160$ goes into 900 five times (5 × 160 = 800), remainder 100.
- Bring down a zero: 1000. $160$ goes into 1000 six times (6 × 160 = 960), remainder 40.
- Bring down a zero: 400. $160$ goes into 400 two times (2 × 160 = 320), remainder 80.
- Bring down a zero: 800. $160$ goes into 800 five times (5 × 160 = 800), remainder 0.

So the decimal is:

$$
0.35625
$$

---

### Final Answer

$$
\boxed{0.35625}
$$

Solution: A terminating decimal can be written in the form $\frac{a}{10^b}$, where $a$ and $b$ are integers. So we try to get a denominator of the form $10^b$: $$\frac{57}{160}=\frac{57}{2^5\cdot5}\cdot\frac{5^4}{5^4}=\frac{57\cdot5^4}{10^5}=\frac{35625}{10^5}=\boxed{.35625}.$$


Explanation:
Okay, so I need to find the last nonzero digit to the right of the decimal point in the decimal expansion of 137 divided by 500. Hmm, let me think about how to approach this.

First, maybe I should just do the division and see what the decimal expansion looks like. Let me try that.

So, 137 divided by 500. Well, 500 goes into 137 zero times. Then, adding a decimal point, we can consider 137.000... divided by 500.

Let me think. 500 times 0.2 is 100, right? So 0.2 times 500 is 100. Then, subtracting 100 from 137, we have 37 left. Bring down a zero, making it 370.

Now, how many times does 500 go into 370? It doesn't, so we put a zero. Then bring down another zero, making it 3700.

Wait, maybe I should do this step by step. Let me write it out properly.

Dividing 137 by 500:

First, since 500 is larger than 137, the result is less than 1. So, we write it as 0. something.

Multiply numerator and denominator by 2 to make the denominator 1000? Wait, 500 times 2 is 1000. So, 137 divided by 500 is the same as (137 * 2) divided by 1000. Let me check that. Yes, because multiplying numerator and denominator by 2 gives (137*2)/1000.

So, 137*2 is 274. Therefore, 274 divided by 1000 is 0.274.

Wait, so the decimal expansion is 0.274? So, the decimal expansion is 0.274, right? So, the digits after the decimal are 2, 7, 4, and then zeros?

But the question is asking for the last nonzero digit to the right of the decimal point. So, if it's 0.274, then the digits after the decimal are 2, 7, 4, and then zeros. So, the last nonzero digit is 4?

But wait, maybe I made a mistake here. Let me verify.

Alternatively, maybe I should do the division step by step. Let me try that again.

Dividing 137 by 500.

So, 500 goes into 137 zero times. So, 0. Then, 500 goes into 1370 two times (since 2*500=1000). Subtract 1000 from 1370, we get 370. Bring down a zero, making it 3700.

Now, 500 goes into 3700 seven times (7*500=3500). Subtract 3500 from 3700, we get 200. Bring down a zero, making it 2000.

500 goes into 2000 four times (4*500=2000). Subtract 2000 from 2000, we get 0.

So, the decimal expansion is 0.274. So, there are no more digits after that. Therefore, the decimal expansion is terminating, right? Because the denominator is 500, which factors into 2^2 * 5^3. Since the prime factors are only 2 and 5, the decimal terminates.

Therefore, the decimal expansion is 0.274, so the digits after the decimal are 2, 7, 4. Therefore, the last nonzero digit is 4.

But wait, the question says "the last nonzero digit to the right of the decimal point". So, if there are no zeros after the decimal, then the last digit is 4.

But maybe I need to check if there are any trailing zeros? For example, if the decimal was 0.274000..., then the last nonzero digit would still be 4. But in this case, since the division ends exactly at 0.274, there are no trailing zeros.

Therefore, the last nonzero digit is 4.

But let me think again. Maybe I made a mistake in the calculation? Let me check 137 divided by 500.

Alternatively, 500 is 5*100, so 137 divided by 500 is equal to (137 divided by 5) divided by 100.

Calculating 137 divided by 5: 5*27 is 135, so 137 divided by 5 is 27.4. Then, divide by 100, so 27.4 divided by 100 is 0.274. Yep, same result.

So, decimal expansion is 0.274. Therefore, the digits after decimal are 2, 7, 4. So, the last nonzero digit is 4.

But maybe the question is trying to trick me? Let me check if there's a different interpretation.

Wait, the question says "the last nonzero digit to the right of the decimal point". So, if there are zeros after the decimal, but before the last nonzero digit, then we have to consider that. But in this case, there are no zeros. The decimal terminates at 4.

Alternatively, maybe I need to consider something else? For example, if the decimal expansion was something like 0.2740... but since it terminates, there are no more digits.

Alternatively, maybe the problem is more complex? Let me think.

Wait, maybe I need to consider the decimal expansion in another way? For example, if the fraction was not terminating, but in this case, it is terminating.

Alternatively, maybe the question is in a different base? No, the question says decimal expansion, so base 10.

Alternatively, maybe I need to consider the decimal expansion of 137/500 in another way? Let me check with another method.

Let me write 137 divided by 500.

Since 500 is 5^3 * 2^2. To convert this into a decimal, we can multiply numerator and denominator by 2^1 to make the denominator 10^3.

So, 137 * 2 = 274, and denominator is 500 * 2 = 1000. Therefore, 274/1000 = 0.274.

Yes, same result.

Therefore, the decimal expansion is 0.274, so the digits after the decimal are 2, 7, 4. Therefore, the last nonzero digit is 4.

But maybe the problem is expecting me to do something else? Let me check if there's a misunderstanding.

Wait, maybe the question is not about the decimal expansion of 137/500, but something else? Let me check again.

No, the question says: "the last nonzero digit to the right of the decimal point in the decimal expansion of 137/500".

So, unless there's a mistake in my calculation, the answer should be 4.

But maybe I need to check with another approach. Let me think.

Another way to find the decimal expansion is to note that 1/500 is 0.002, so 137/500 is 137 * 0.002.

Calculating 137 * 0.002:

First, 100 * 0.002 = 0.2

30 * 0.002 = 0.06

7 * 0.002 = 0.014

Adding them up: 0.2 + 0.06 = 0.26; 0.26 + 0.014 = 0.274.

Same result.

Therefore, decimal expansion is 0.274.

Therefore, the digits after decimal are 2, 7, 4.

So, the last nonzero digit is 4.

But maybe the problem is in another context? For example, if the decimal expansion was longer? But no, since 500 divides into 10^3, so the decimal terminates at three decimal places.

Alternatively, maybe the question is a trick question? For example, if you consider that after the decimal, there are digits, but maybe there's a zero? But no, 0.274 is exact.

Alternatively, maybe I need to consider the decimal expansion in a different way? For example, if you have 137 divided by 500, and you want to write it as a decimal, but maybe there's some rounding? But the question says "decimal expansion", which is exact.

Alternatively, maybe the problem is from a source where they consider something else?

Wait, let me check with another method. Let me do the division manually again.

Dividing 137 by 500.

So, 500 goes into 137 zero times. Put decimal point.

500 goes into 1370 two times (2*500=1000). Subtract 1000 from 1370, get 370.

Bring down a zero: 3700.

500 goes into 3700 seven times (7*500=3500). Subtract 3500 from 3700, get 200.

Bring down a zero: 2000.

500 goes into 2000 four times (4*500=2000). Subtract 2000, get 0.

So, the decimal is 0.274.

Therefore, the decimal expansion is exactly 0.274.

Therefore, the digits after the decimal are 2, 7, 4.

So, the last nonzero digit is 4.

Therefore, the answer is 4.

But maybe the problem is expecting me to think about something else? For example, if there was a repeating decimal? But no, since denominator is 500, which factors into 2^2 * 5^3, so it's a terminating decimal.

Alternatively, maybe the question is in a different language or something? No, the question is in English.

Alternatively, maybe I need to consider the decimal expansion of 137/500 in another way? For example, if I write it as 0.274, then the digits after decimal are 2, 7, 4. So, the last nonzero digit is 4.

Alternatively, maybe the problem is from a math competition where sometimes they have tricks? For example, maybe they consider something else?

Wait, let me check with another approach. Let me think about the decimal expansion.

Since 1/500 = 0.002, as I said before. Then 137/500 = 137 * 0.002.

But 137 * 0.002 = 0.274.

Alternatively, maybe I can think of 137 divided by 500 as follows:

Divide numerator and denominator by 5: 137/500 = 27.4/100.

Which is 0.274.

Same result.

Alternatively, maybe I can think of 137 divided by 500 as multiplying by 1/500.

But again, same result.

Alternatively, maybe I can think of 137 divided by 500 as 137 divided by 5 divided by 100.

Which is 27.4 divided by 100, which is 0.274.

Same result.

Therefore, I think my answer is correct.

But just to be thorough, let me check with another method. Let me convert the fraction to a decimal by multiplying numerator and denominator by something to get denominator as a power of 10.

Since denominator is 500, which is 5^3 * 2^2. To make it a power of 10, we need to multiply numerator and denominator by 2^1, so that denominator becomes 5^3 * 2^3 = 10^3.

So, multiplying numerator and denominator by 2:

137 * 2 = 274

Denominator: 500 * 2 = 1000

Therefore, 274/1000 = 0.274.

Same result.

Therefore, decimal expansion is 0.274.

Therefore, the last nonzero digit to the right of the decimal is 4.

Therefore, the answer is 4.

But just to make sure, let me check with a calculator.

Well, I don't have a calculator here, but I can do the multiplication.

Let me check 500 * 0.274.

First, 500 * 0.2 = 100

500 * 0.07 = 35

500 * 0.004 = 2

Adding them up: 100 + 35 = 135; 135 + 2 = 137.

Yes! So, 500 * 0.274 = 137. Therefore, 137 / 500 = 0.274.

Therefore, the decimal expansion is exactly 0.274.

Therefore, the digits after the decimal are 2, 7, 4.

So, the last nonzero digit is 4.

Therefore, the answer is 4.

But maybe the problem is expecting me to consider something else? For example, if there was a decimal expansion with more digits? But no, since it's terminating.

Alternatively, maybe the question is in a different form? For example, if the fraction was 137/500, but written in some other way?

Alternatively, maybe the question is a trick question where the decimal expansion is 0.274... with some more digits? But no, since 500 divides into 1000 exactly, so the decimal terminates.

Alternatively, maybe I need to consider the decimal expansion in another base? But the question says decimal expansion, so base 10.

Therefore, I think my answer is correct.

Therefore, the last nonzero digit to the right of the decimal point in the decimal expansion of 137/500 is 4.

**Final Answer**
The last nonzero digit to the right of the decimal point is \boxed{4}.

To determine the **last nonzero digit to the right of the decimal point** in the decimal expansion of $\frac{137}{500}$, we begin by analyzing the structure of the fraction and its decimal representation.

---

### Step 1: Understand the Nature of the Fraction

The denominator, 500, can be factored as:

$$
500 = 2^2 \times 5^3
$$

Since the denominator only contains the prime factors 2 and 5, the decimal expansion of the fraction will **terminate**. This is a key property of fractions whose denominators are of the form $2^a \times 5^b$, and such fractions always have terminating decimal expansions.

---

### Step 2: Convert the Fraction to a Decimal

To convert $\frac{137}{500}$ into a decimal, we can multiply both numerator and denominator by 2 to make the denominator a power of 10:

$$
\frac{137}{500} = \frac{137 \times 2}{500 \times 2} = \frac{274}{1000}
$$

Now, we can write this as:

$$
\frac{274}{1000} = 0.274
$$

So, the decimal expansion of $\frac{137}{500}$ is **exactly** $0.274$, with no further digits.

---

### Step 3: Identify the Last Nonzero Digit

Looking at the decimal expansion $0.274$, the digits to the right of the decimal point are:

- 2
- 7
- 4

All of these digits are **nonzero**, and since the decimal terminates at the third digit, there are **no trailing zeros** after the 4.

Thus, the **last nonzero digit to the right of the decimal point** is:

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the last nonzero digit to the right of the decimal point in the decimal expansion of 137 divided by 500. Hmm, let me think about how to approach this.

First, maybe I should just do the division and see what the decimal expansion looks like. Let me try that.

So, 137 divided by 500. Well, 500 goes into 137 zero times. Then, adding a decimal point, we can consider 137.000... divided by 500.

Let me think. 500 times 0.2 is 100, right? So 0.2 times 500 is 100. Then, subtracting 100 from 137, we have 37 left. Bring down a zero, making it 370.

Now, how many times does 500 go into 370? It doesn't, so we put a zero. Then bring down another zero, making it 3700.

Wait, maybe I should do this step by step. Let me write it out properly.

Dividing 137 by 500:

First, since 500 is larger than 137, the result is less than 1. So, we write it as 0. something.

Multiply numerator and denominator by 2 to make the denominator 1000? Wait, 500 times 2 is 1000. So, 137 divided by 500 is the same as (137 * 2) divided by 1000. Let me check that. Yes, because multiplying numerator and denominator by 2 gives (137*2)/1000.

So, 137*2 is 274. Therefore, 274 divided by 1000 is 0.274.

Wait, so the decimal expansion is 0.274? So, the decimal expansion is 0.274, right? So, the digits after the decimal are 2, 7, 4, and then zeros?

But the question is asking for the last nonzero digit to the right of the decimal point. So, if it's 0.274, then the digits after the decimal are 2, 7, 4, and then zeros. So, the last nonzero digit is 4?

But wait, maybe I made a mistake here. Let me verify.

Alternatively, maybe I should do the division step by step. Let me try that again.

Dividing 137 by 500.

So, 500 goes into 137 zero times. So, 0. Then, 500 goes into 1370 two times (since 2*500=1000). Subtract 1000 from 1370, we get 370. Bring down a zero, making it 3700.

Now, 500 goes into 3700 seven times (7*500=3500). Subtract 3500 from 3700, we get 200. Bring down a zero, making it 2000.

500 goes into 2000 four times (4*500=2000). Subtract 2000 from 2000, we get 0.

So, the decimal expansion is 0.274. So, there are no more digits after that. Therefore, the decimal expansion is terminating, right? Because the denominator is 500, which factors into 2^2 * 5^3. Since the prime factors are only 2 and 5, the decimal terminates.

Therefore, the decimal expansion is 0.274, so the digits after the decimal are 2, 7, 4. Therefore, the last nonzero digit is 4.

But wait, the question says "the last nonzero digit to the right of the decimal point". So, if there are no zeros after the decimal, then the last digit is 4.

But maybe I need to check if there are any trailing zeros? For example, if the decimal was 0.274000..., then the last nonzero digit would still be 4. But in this case, since the division ends exactly at 0.274, there are no trailing zeros.

Therefore, the last nonzero digit is 4.

But let me think again. Maybe I made a mistake in the calculation? Let me check 137 divided by 500.

Alternatively, 500 is 5*100, so 137 divided by 500 is equal to (137 divided by 5) divided by 100.

Calculating 137 divided by 5: 5*27 is 135, so 137 divided by 5 is 27.4. Then, divide by 100, so 27.4 divided by 100 is 0.274. Yep, same result.

So, decimal expansion is 0.274. Therefore, the digits after decimal are 2, 7, 4. So, the last nonzero digit is 4.

But maybe the question is trying to trick me? Let me check if there's a different interpretation.

Wait, the question says "the last nonzero digit to the right of the decimal point". So, if there are zeros after the decimal, but before the last nonzero digit, then we have to consider that. But in this case, there are no zeros. The decimal terminates at 4.

Alternatively, maybe I need to consider something else? For example, if the decimal expansion was something like 0.2740... but since it terminates, there are no more digits.

Alternatively, maybe the problem is more complex? Let me think.

Wait, maybe I need to consider the decimal expansion in another way? For example, if the fraction was not terminating, but in this case, it is terminating.

Alternatively, maybe the question is in a different base? No, the question says decimal expansion, so base 10.

Alternatively, maybe I need to consider the decimal expansion of 137/500 in another way? Let me check with another method.

Let me write 137 divided by 500.

Since 500 is 5^3 * 2^2. To convert this into a decimal, we can multiply numerator and denominator by 2^1 to make the denominator 10^3.

So, 137 * 2 = 274, and denominator is 500 * 2 = 1000. Therefore, 274/1000 = 0.274.

Yes, same result.

Therefore, the decimal expansion is 0.274, so the digits after the decimal are 2, 7, 4. Therefore, the last nonzero digit is 4.

But maybe the problem is expecting me to do something else? Let me check if there's a misunderstanding.

Wait, maybe the question is not about the decimal expansion of 137/500, but something else? Let me check again.

No, the question says: "the last nonzero digit to the right of the decimal point in the decimal expansion of 137/500".

So, unless there's a mistake in my calculation, the answer should be 4.

But maybe I need to check with another approach. Let me think.

Another way to find the decimal expansion is to note that 1/500 is 0.002, so 137/500 is 137 * 0.002.

Calculating 137 * 0.002:

First, 100 * 0.002 = 0.2

30 * 0.002 = 0.06

7 * 0.002 = 0.014

Adding them up: 0.2 + 0.06 = 0.26; 0.26 + 0.014 = 0.274.

Same result.

Therefore, decimal expansion is 0.274.

Therefore, the digits after decimal are 2, 7, 4.

So, the last nonzero digit is 4.

But maybe the problem is in another context? For example, if the decimal expansion was longer? But no, since 500 divides into 10^3, so the decimal terminates at three decimal places.

Alternatively, maybe the question is a trick question? For example, if you consider that after the decimal, there are digits, but maybe there's a zero? But no, 0.274 is exact.

Alternatively, maybe I need to consider the decimal expansion in a different way? For example, if you have 137 divided by 500, and you want to write it as a decimal, but maybe there's some rounding? But the question says "decimal expansion", which is exact.

Alternatively, maybe the problem is from a source where they consider something else?

Wait, let me check with another method. Let me do the division manually again.

Dividing 137 by 500.

So, 500 goes into 137 zero times. Put decimal point.

500 goes into 1370 two times (2*500=1000). Subtract 1000 from 1370, get 370.

Bring down a zero: 3700.

500 goes into 3700 seven times (7*500=3500). Subtract 3500 from 3700, get 200.

Bring down a zero: 2000.

500 goes into 2000 four times (4*500=2000). Subtract 2000, get 0.

So, the decimal is 0.274.

Therefore, the decimal expansion is exactly 0.274.

Therefore, the digits after the decimal are 2, 7, 4.

So, the last nonzero digit is 4.

Therefore, the answer is 4.

But maybe the problem is expecting me to think about something else? For example, if there was a repeating decimal? But no, since denominator is 500, which factors into 2^2 * 5^3, so it's a terminating decimal.

Alternatively, maybe the question is in a different language or something? No, the question is in English.

Alternatively, maybe I need to consider the decimal expansion of 137/500 in another way? For example, if I write it as 0.274, then the digits after decimal are 2, 7, 4. So, the last nonzero digit is 4.

Alternatively, maybe the problem is from a math competition where sometimes they have tricks? For example, maybe they consider something else?

Wait, let me check with another approach. Let me think about the decimal expansion.

Since 1/500 = 0.002, as I said before. Then 137/500 = 137 * 0.002.

But 137 * 0.002 = 0.274.

Alternatively, maybe I can think of 137 divided by 500 as follows:

Divide numerator and denominator by 5: 137/500 = 27.4/100.

Which is 0.274.

Same result.

Alternatively, maybe I can think of 137 divided by 500 as multiplying by 1/500.

But again, same result.

Alternatively, maybe I can think of 137 divided by 500 as 137 divided by 5 divided by 100.

Which is 27.4 divided by 100, which is 0.274.

Same result.

Therefore, I think my answer is correct.

But just to be thorough, let me check with another method. Let me convert the fraction to a decimal by multiplying numerator and denominator by something to get denominator as a power of 10.

Since denominator is 500, which is 5^3 * 2^2. To make it a power of 10, we need to multiply numerator and denominator by 2^1, so that denominator becomes 5^3 * 2^3 = 10^3.

So, multiplying numerator and denominator by 2:

137 * 2 = 274

Denominator: 500 * 2 = 1000

Therefore, 274/1000 = 0.274.

Same result.

Therefore, decimal expansion is 0.274.

Therefore, the last nonzero digit to the right of the decimal is 4.

Therefore, the answer is 4.

But just to make sure, let me check with a calculator.

Well, I don't have a calculator here, but I can do the multiplication.

Let me check 500 * 0.274.

First, 500 * 0.2 = 100

500 * 0.07 = 35

500 * 0.004 = 2

Adding them up: 100 + 35 = 135; 135 + 2 = 137.

Yes! So, 500 * 0.274 = 137. Therefore, 137 / 500 = 0.274.

Therefore, the decimal expansion is exactly 0.274.

Therefore, the digits after the decimal are 2, 7, 4.

So, the last nonzero digit is 4.

Therefore, the answer is 4.

But maybe the problem is expecting me to consider something else? For example, if there was a decimal expansion with more digits? But no, since it's terminating.

Alternatively, maybe the question is in a different form? For example, if the fraction was 137/500, but written in some other way?

Alternatively, maybe the question is a trick question where the decimal expansion is 0.274... with some more digits? But no, since 500 divides into 1000 exactly, so the decimal terminates.

Alternatively, maybe I need to consider the decimal expansion in another base? But the question says decimal expansion, so base 10.

Therefore, I think my answer is correct.

Therefore, the last nonzero digit to the right of the decimal point in the decimal expansion of 137/500 is 4.

**Final Answer**
The last nonzero digit to the right of the decimal point is \boxed{4}.

To determine the **last nonzero digit to the right of the decimal point** in the decimal expansion of $\frac{137}{500}$, we begin by analyzing the structure of the fraction and its decimal representation.

---

### Step 1: Understand the Nature of the Fraction

The denominator, 500, can be factored as:

$$
500 = 2^2 \times 5^3
$$

Since the denominator only contains the prime factors 2 and 5, the decimal expansion of the fraction will **terminate**. This is a key property of fractions whose denominators are of the form $2^a \times 5^b$, and such fractions always have terminating decimal expansions.

---

### Step 2: Convert the Fraction to a Decimal

To convert $\frac{137}{500}$ into a decimal, we can multiply both numerator and denominator by 2 to make the denominator a power of 10:

$$
\frac{137}{500} = \frac{137 \times 2}{500 \times 2} = \frac{274}{1000}
$$

Now, we can write this as:

$$
\frac{274}{1000} = 0.274
$$

So, the decimal expansion of $\frac{137}{500}$ is **exactly** $0.274$, with no further digits.

---

### Step 3: Identify the Last Nonzero Digit

Looking at the decimal expansion $0.274$, the digits to the right of the decimal point are:

- 2
- 7
- 4

All of these digits are **nonzero**, and since the decimal terminates at the third digit, there are **no trailing zeros** after the 4.

Thus, the **last nonzero digit to the right of the decimal point** is:

$$
\boxed{4}
$$

Solution: We can rewrite $\frac{137}{500}$ in the form $\frac{274}{1000}$, so $\frac{137}{500} = \frac{274}{1000} = 0.274$ and the last nonzero digit is $\boxed{4}$.