Explanation:
Okay, so I need to find an integer n between 0 and 17 (inclusive) such that n is congruent to -11213141 modulo 18. Hmm, let me think about how to approach this.

First, I remember that when dealing with congruences, especially negative numbers, it's helpful to convert them into positive equivalents by adding multiples of the modulus. Since the modulus here is 18, I can add 18 enough times to the negative number until it becomes positive. But dealing with such a large number as -11213141 directly might be tedious. Maybe there's a smarter way?

Alternatively, I recall that for any integer a and modulus m, a ≡ b mod m if and only if a - b is divisible by m. But I'm not sure if that helps directly here. Wait, maybe I can use properties of modular arithmetic to simplify the problem.

Another thought: since we're working modulo 18, perhaps I can find the remainder when 11213141 is divided by 18, and then take the negative of that remainder, adjusting it to be within the 0 to 17 range. That might work. Let me try that.

But how do I compute 11213141 mod 18? Dividing such a large number by 18 manually would take a while. Maybe there's a shortcut? I remember that there's a rule for divisibility by 9 and 2, which are factors of 18. Since 18 = 2 * 9, maybe I can use that?

Wait, but actually, for modulus 18, maybe I can use the fact that 10 ≡ -8 mod 18? Or perhaps break down the number digit by digit? Let me think.

Alternatively, I remember that for modulus 9, the remainder of a number is equal to the sum of its digits modulo 9. Similarly, for modulus 2, it's just the last digit. But since 18 is 2*9, maybe I can use Chinese Remainder Theorem? Wait, but maybe that complicates things. Let me try the digit sum approach for modulus 9 first.

Let me compute the sum of the digits of 11213141. Let's see:

Digits are 1, 1, 2, 1, 3, 1, 4, 1.

Adding them up: 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1. Let me compute that step by step:

Start with 1 + 1 = 2

2 + 2 = 4

4 + 1 = 5

5 + 3 = 8

8 + 1 = 9

9 + 4 = 13

13 + 1 = 14.

So the sum of the digits is 14. Therefore, 11213141 ≡ 14 mod 9. But since 14 divided by 9 is 1 with remainder 5, so 14 ≡ 5 mod 9. Therefore, 11213141 ≡ 5 mod 9.

But we need mod 18. Hmm. How can I get mod 18 from mod 9? Well, maybe I can find the remainder mod 2 as well. Since 18 is 2*9, and 2 and 9 are coprime.

So, if I can find 11213141 mod 2 and mod 9, then I can use Chinese Remainder Theorem to find mod 18.

First, mod 2: the number is 11213141. The last digit is 1, which is odd, so 11213141 ≡ 1 mod 2.

Earlier, we found that 11213141 ≡ 5 mod 9.

So, we need to find a number x such that:

x ≡ 5 mod 9

x ≡ 1 mod 2

And then x will be congruent to 11213141 mod 18. Then, since we need -11213141 mod 18, that would be -x mod 18.

So let me solve for x first. Let me find numbers congruent to 5 mod 9 and 1 mod 2. Let me list numbers congruent to 5 mod 9: 5, 14, 23, 32, 41, 50, 59, 68, 77, 86, 95, etc. Now check which of these are congruent to 1 mod 2.

Looking at the numbers:

5 is odd (5 mod 2 = 1) → yes. So x ≡ 5 mod 18? Wait, 5 mod 2 is 1, so 5 satisfies both congruences. Therefore, x ≡ 5 mod 18. Therefore, 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18.

But -5 mod 18 is the same as 18 - 5 = 13. Therefore, n ≡ 13 mod 18. Since 0 ≤ n < 18, n = 13.

Wait, but let me check this again. Because if 11213141 ≡ 5 mod 18, then -11213141 ≡ -5 mod 18. Then adding 18 to -5 gives 13. So yes, n = 13.

But let me verify this with another method to be sure. Maybe compute 11213141 mod 18 directly?

Alternatively, since 18 is 9*2, and we already have mod 9 and mod 2, but maybe there's another way. Alternatively, maybe compute the number mod 18 by breaking it down.

Alternatively, note that 10 ≡ 10 mod 18, 10^2 ≡ 100 ≡ 100 - 5*18 = 100 - 90 = 10 mod 18. Wait, 10^2 ≡ 10 mod 18? Let me check:

10^1 = 10 mod 18

10^2 = 10*10 = 100. 100 divided by 18: 18*5=90, 100-90=10. So yes, 10^2 ≡ 10 mod 18. Similarly, 10^3 = 10^2 * 10 ≡ 10*10 = 100 ≡ 10 mod 18. So in general, 10^k ≡ 10 mod 18 for k ≥ 1.

Wait, that seems interesting. So if that's the case, then any number can be represented as a sum of digits multiplied by powers of 10. So for example, the number 11213141 can be written as:

1*10^7 + 1*10^6 + 2*10^5 + 1*10^4 + 3*10^3 + 1*10^2 + 4*10^1 + 1*10^0.

But since each 10^k ≡ 10 mod 18 for k ≥ 1, except for 10^0 = 1. Therefore, each term with exponent ≥1 is congruent to 10 mod 18. Let me check:

For example, 10^1 ≡ 10, 10^2 ≡ 10, 10^3 ≡ 10, etc. So, for the number 11213141, which is:

Digits: positions from right (starting at 0):

Position 7: 1

Position 6: 1

Position 5: 2

Position 4: 1

Position 3: 3

Position 2: 1

Position 1: 4

Position 0: 1

Wait, actually, the positions are from right to left, but when we write the number, the leftmost digit is the highest power. Let me clarify:

The number is 11,213,141. Let me write it as:

Digit positions (from right, starting at 0):

Position 7: 1 (ten millions place)

Position 6: 1 (millions)

Position 5: 2 (hundred thousands)

Position 4: 1 (ten thousands)

Position 3: 3 (thousands)

Position 2: 1 (hundreds)

Position 1: 4 (tens)

Position 0: 1 (units)

So, the value is:

1*10^7 + 1*10^6 + 2*10^5 + 1*10^4 + 3*10^3 + 1*10^2 + 4*10^1 + 1*10^0.

As per earlier, 10^k ≡ 10 mod 18 for k ≥ 1. Therefore, each term with exponent ≥1 is 10 mod 18. So:

Let me compute each term mod 18:

For the first term, 1*10^7 ≡ 1*10 mod 18 = 10

Second term: 1*10^6 ≡ 1*10 mod 18 = 10

Third term: 2*10^5 ≡ 2*10 = 20 mod 18 = 2

Fourth term: 1*10^4 ≡ 1*10 = 10

Fifth term: 3*10^3 ≡ 3*10 = 30 mod 18. 30 - 18 = 12, so 12

Sixth term: 1*10^2 ≡ 1*10 = 10

Seventh term: 4*10^1 ≡ 4*10 = 40 mod 18. 40 - 2*18 = 40 - 36 = 4

Eighth term: 1*10^0 = 1

Now, adding all these up:

Start with 10 (from first term)

+10 = 20

+2 = 22

+10 = 32

+12 = 44

+10 = 54

+4 = 58

+1 = 59

So total is 59 mod 18. Now compute 59 divided by 18. 18*3=54, so 59 - 54 = 5. Therefore, 59 ≡ 5 mod 18.

So that confirms that 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18. Then, -5 + 18 = 13, so -5 ≡ 13 mod 18. Therefore, n = 13.

But let me check once more with another approach. Maybe compute the number mod 18 by using the fact that 10 ≡ 10 mod 18, 10^2 ≡ 10 mod 18, etc., but maybe there's a pattern. Alternatively, since 10 ≡ -8 mod 18, maybe that helps? Let me try that.

If 10 ≡ -8 mod 18, then 10^1 ≡ -8, 10^2 ≡ (-8)*(-8) = 64 mod 18. 64 divided by 18 is 3*18=54, 64-54=10. So 10^2 ≡ 10 mod 18. Which matches the previous result. Then 10^3 ≡ 10*10 ≡ 10*10 = 100 ≡ 10 mod 18. So same as before. So regardless of the exponent, 10^k ≡ 10 mod 18 for k ≥ 1.

Therefore, the same calculation applies. So the total is 5 mod 18. Therefore, -5 mod 18 is 13.

Alternatively, maybe I can compute the original number mod 18 by breaking it down into smaller parts. For example, compute 11213141 divided by 18, but that would be tedious. Alternatively, use the fact that 18*622,952 = 11,213,136. Let me check that.

Wait, 18*600,000 = 10,800,000. Then 18*22,952 = ?

Wait, maybe this is too time-consuming. Alternatively, since we already have two methods giving 5 mod 18, which seems consistent, maybe I can trust that.

But just to be thorough, let me try another way. Let me compute 11213141 mod 18 by using the fact that 18*622,952 = 11,213,136. Then 11,213,141 - 11,213,136 = 5. Therefore, 11213141 ≡ 5 mod 18. Yes! So that confirms it. Therefore, -11213141 ≡ -5 mod 18, which is 13 mod 18.

Therefore, the integer n is 13.

But just to make sure, let me check with another method. Let me compute the number mod 18 by using the fact that 10 ≡ 10 mod 18, so we can compute the number as follows:

Starting from the rightmost digit, we can compute the value step by step. Let me try that.

The number is 11213141. Let me write it as digits: 1, 1, 2, 1, 3, 1, 4, 1. From left to right. Wait, but when computing mod 18, maybe it's easier to process from right to left? Let me think.

Alternatively, here's a method: take the number and repeatedly take mod 18 as you go. For example, starting with 0, then for each digit, compute (current_value * 10 + digit) mod 18.

Let me try that. Let's take the number 11213141. Let's process each digit from left to right:

Start with 0.

First digit: 1. Compute (0 * 10 + 1) mod 18 = 1 mod 18 = 1.

Second digit: 1. Compute (1 * 10 + 1) = 11 mod 18 = 11.

Third digit: 2. Compute (11 * 10 + 2) = 112 mod 18. Let me compute 112 / 18: 18*6=108, 112-108=4. So 4.

Fourth digit: 1. Compute (4 * 10 + 1) = 41 mod 18. 18*2=36, 41-36=5. So 5.

Fifth digit: 3. Compute (5 * 10 + 3) = 53 mod 18. 18*2=36, 53-36=17. 17 mod 18 is 17.

Sixth digit: 1. Compute (17 * 10 + 1) = 171 mod 18. 18*9=162, 171-162=9. So 9.

Seventh digit: 4. Compute (9 * 10 + 4) = 94 mod 18. 18*5=90, 94-90=4. So 4.

Eighth digit: 1. Compute (4 * 10 + 1) = 41 mod 18. As before, 41 - 36 = 5. So 5.

Wait, so according to this method, the result is 5 mod 18. Which matches our previous results. Therefore, 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18. Which is 13 mod 18.

Therefore, n = 13.

But just to be absolutely sure, let me check with another approach. Let me compute 11213141 divided by 18 and find the remainder.

But doing this manually would be time-consuming, but maybe I can use the fact that 18*600,000 = 10,800,000. Then subtract that from 11,213,141:

11,213,141 - 10,800,000 = 413,141.

Now compute 413,141 divided by 18. Let me find how many times 18 goes into 413,141.

First, 18*20,000 = 360,000. Subtract that from 413,141: 413,141 - 360,000 = 53,141.

Now, 18*3,000 = 54,000. That's more than 53,141. So try 18*2,950 = 18*(3,000 - 50) = 54,000 - 900 = 53,100.

Subtract 53,100 from 53,141: 53,141 - 53,100 = 41.

So total so far is 600,000 + 20,000 + 2,950 = 622,950. And the remainder is 41.

Now, 41 divided by 18 is 2*18=36, remainder 5. Therefore, total remainder is 5. Therefore, 11,213,141 divided by 18 gives a remainder of 5. Therefore, 11213141 ≡ 5 mod 18.

Therefore, -11213141 ≡ -5 mod 18. Which is 13 mod 18.

Therefore, n = 13.

So all methods lead to the same conclusion. Therefore, I can be confident that the answer is 13.

But just to recap, the steps were:

1. Recognize that finding -11213141 mod 18 is equivalent to finding (-11213141) mod 18, which can be converted to a positive number by adding 18 until it's within 0-17.

2. Instead of dealing with the large number directly, use properties of modular arithmetic to simplify.

3. Noted that modulus 18 can be broken down into modulus 9 and modulus 2, then used Chinese Remainder Theorem to find the value mod 18.

4. Verified using multiple methods: digit sum for modulus 9, breaking down the number into powers of 10, processing digits step by step, and direct division to find the remainder.

All methods confirmed that 11213141 ≡ 5 mod 18, so -11213141 ≡ -5 ≡ 13 mod 18.

Therefore, the integer n is 13.

**Final Answer**
The integer $ n $ is \boxed{13}.

To solve the problem of finding an integer $ n $ such that:

$$
0 \leq n < 18 \quad \text{and} \quad n \equiv -11213141 \pmod{18}
$$

we aim to compute the value of $ -11213141 \mod 18 $, and then ensure the result lies in the range $ [0, 17] $.

---

### Step 1: Understand the Problem

We are given a large negative number, $ -11213141 $, and asked to find its residue modulo 18. Since the modulus is 18, the result must be an integer between 0 and 17 (inclusive). That is, we are looking for:

$$
n \equiv -11213141 \pmod{18}
$$

This is equivalent to:

$$
n \equiv (-11213141 \mod 18) \pmod{18}
$$

So, the key is to compute $ -11213141 \mod 18 $, and then adjust it to the range $ [0, 17] $ if necessary.

---

### Step 2: Compute $ 11213141 \mod 18 $

Rather than directly computing $ -11213141 \mod 18 $, we can first compute $ 11213141 \mod 18 $, and then take the negative of that result modulo 18.

#### Method: Use the fact that $ 10^k \equiv 10 \mod 18 $ for $ k \geq 1 $

We can break the number into its digits and use the fact that:

$$
10^k \equiv 10 \mod 18 \quad \text{for all } k \geq 1
$$

So, for the number $ 11213141 $, we can write it as:

$$
1 \cdot 10^7 + 1 \cdot 10^6 + 2 \cdot 10^5 + 1 \cdot 10^4 + 3 \cdot 10^3 + 1 \cdot 10^2 + 4 \cdot 10^1 + 1 \cdot 10^0
$$

Now, since $ 10^k \equiv 10 \mod 18 $ for $ k \geq 1 $, and $ 10^0 = 1 $, we can compute:

$$
\begin{align*}
1 \cdot 10 &\equiv 10 \\
1 \cdot 10 &\equiv 10 \\
2 \cdot 10 &\equiv 20 \equiv 2 \mod 18 \\
1 \cdot 10 &\equiv 10 \\
3 \cdot 10 &\equiv 30 \equiv 12 \mod 18 \\
1 \cdot 10 &\equiv 10 \\
4 \cdot 10 &\equiv 40 \equiv 4 \mod 18 \\
1 \cdot 1 &\equiv 1 \\
\end{align*}
$$

Now sum these values:

$$
10 + 10 + 2 + 10 + 12 + 10 + 4 + 1 = 59
$$

Now compute $ 59 \mod 18 $:

$$
59 \div 18 = 3 \text{ remainder } 5 \Rightarrow 59 \equiv 5 \mod 18
$$

So:

$$
11213141 \equiv 5 \mod 18
$$

---

### Step 3: Compute $ -11213141 \mod 18 $

Now that we know:

$$
11213141 \equiv 5 \mod 18
$$

Then:

$$
-11213141 \equiv -5 \mod 18
$$

To convert this to a positive value in the range $ [0, 17] $, we add 18:

$$
-5 + 18 = 13
$$

So:

$$
-11213141 \equiv 13 \mod 18
$$

---

### Final Answer

$$
\boxed{13}
$$

[Assistant]
Explanation:
Okay, so I need to find an integer n between 0 and 17 (inclusive) such that n is congruent to -11213141 modulo 18. Hmm, let me think about how to approach this.

First, I remember that when dealing with congruences, especially negative numbers, it's helpful to convert them into positive equivalents by adding multiples of the modulus. Since the modulus here is 18, I can add 18 enough times to the negative number until it becomes positive. But dealing with such a large number as -11213141 directly might be tedious. Maybe there's a smarter way?

Alternatively, I recall that for any integer a and modulus m, a ≡ b mod m if and only if a - b is divisible by m. But I'm not sure if that helps directly here. Wait, maybe I can use properties of modular arithmetic to simplify the problem.

Another thought: since we're working modulo 18, perhaps I can find the remainder when 11213141 is divided by 18, and then take the negative of that remainder, adjusting it to be within the 0 to 17 range. That might work. Let me try that.

But how do I compute 11213141 mod 18? Dividing such a large number by 18 manually would take a while. Maybe there's a shortcut? I remember that there's a rule for divisibility by 9 and 2, which are factors of 18. Since 18 = 2 * 9, maybe I can use that?

Wait, but actually, for modulus 18, maybe I can use the fact that 10 ≡ -8 mod 18? Or perhaps break down the number digit by digit? Let me think.

Alternatively, I remember that for modulus 9, the remainder of a number is equal to the sum of its digits modulo 9. Similarly, for modulus 2, it's just the last digit. But since 18 is 2*9, maybe I can use Chinese Remainder Theorem? Wait, but maybe that complicates things. Let me try the digit sum approach for modulus 9 first.

Let me compute the sum of the digits of 11213141. Let's see:

Digits are 1, 1, 2, 1, 3, 1, 4, 1.

Adding them up: 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1. Let me compute that step by step:

Start with 1 + 1 = 2

2 + 2 = 4

4 + 1 = 5

5 + 3 = 8

8 + 1 = 9

9 + 4 = 13

13 + 1 = 14.

So the sum of the digits is 14. Therefore, 11213141 ≡ 14 mod 9. But since 14 divided by 9 is 1 with remainder 5, so 14 ≡ 5 mod 9. Therefore, 11213141 ≡ 5 mod 9.

But we need mod 18. Hmm. How can I get mod 18 from mod 9? Well, maybe I can find the remainder mod 2 as well. Since 18 is 2*9, and 2 and 9 are coprime.

So, if I can find 11213141 mod 2 and mod 9, then I can use Chinese Remainder Theorem to find mod 18.

First, mod 2: the number is 11213141. The last digit is 1, which is odd, so 11213141 ≡ 1 mod 2.

Earlier, we found that 11213141 ≡ 5 mod 9.

So, we need to find a number x such that:

x ≡ 5 mod 9

x ≡ 1 mod 2

And then x will be congruent to 11213141 mod 18. Then, since we need -11213141 mod 18, that would be -x mod 18.

So let me solve for x first. Let me find numbers congruent to 5 mod 9 and 1 mod 2. Let me list numbers congruent to 5 mod 9: 5, 14, 23, 32, 41, 50, 59, 68, 77, 86, 95, etc. Now check which of these are congruent to 1 mod 2.

Looking at the numbers:

5 is odd (5 mod 2 = 1) → yes. So x ≡ 5 mod 18? Wait, 5 mod 2 is 1, so 5 satisfies both congruences. Therefore, x ≡ 5 mod 18. Therefore, 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18.

But -5 mod 18 is the same as 18 - 5 = 13. Therefore, n ≡ 13 mod 18. Since 0 ≤ n < 18, n = 13.

Wait, but let me check this again. Because if 11213141 ≡ 5 mod 18, then -11213141 ≡ -5 mod 18. Then adding 18 to -5 gives 13. So yes, n = 13.

But let me verify this with another method to be sure. Maybe compute 11213141 mod 18 directly?

Alternatively, since 18 is 9*2, and we already have mod 9 and mod 2, but maybe there's another way. Alternatively, maybe compute the number mod 18 by breaking it down.

Alternatively, note that 10 ≡ 10 mod 18, 10^2 ≡ 100 ≡ 100 - 5*18 = 100 - 90 = 10 mod 18. Wait, 10^2 ≡ 10 mod 18? Let me check:

10^1 = 10 mod 18

10^2 = 10*10 = 100. 100 divided by 18: 18*5=90, 100-90=10. So yes, 10^2 ≡ 10 mod 18. Similarly, 10^3 = 10^2 * 10 ≡ 10*10 = 100 ≡ 10 mod 18. So in general, 10^k ≡ 10 mod 18 for k ≥ 1.

Wait, that seems interesting. So if that's the case, then any number can be represented as a sum of digits multiplied by powers of 10. So for example, the number 11213141 can be written as:

1*10^7 + 1*10^6 + 2*10^5 + 1*10^4 + 3*10^3 + 1*10^2 + 4*10^1 + 1*10^0.

But since each 10^k ≡ 10 mod 18 for k ≥ 1, except for 10^0 = 1. Therefore, each term with exponent ≥1 is congruent to 10 mod 18. Let me check:

For example, 10^1 ≡ 10, 10^2 ≡ 10, 10^3 ≡ 10, etc. So, for the number 11213141, which is:

Digits: positions from right (starting at 0):

Position 7: 1

Position 6: 1

Position 5: 2

Position 4: 1

Position 3: 3

Position 2: 1

Position 1: 4

Position 0: 1

Wait, actually, the positions are from right to left, but when we write the number, the leftmost digit is the highest power. Let me clarify:

The number is 11,213,141. Let me write it as:

Digit positions (from right, starting at 0):

Position 7: 1 (ten millions place)

Position 6: 1 (millions)

Position 5: 2 (hundred thousands)

Position 4: 1 (ten thousands)

Position 3: 3 (thousands)

Position 2: 1 (hundreds)

Position 1: 4 (tens)

Position 0: 1 (units)

So, the value is:

1*10^7 + 1*10^6 + 2*10^5 + 1*10^4 + 3*10^3 + 1*10^2 + 4*10^1 + 1*10^0.

As per earlier, 10^k ≡ 10 mod 18 for k ≥ 1. Therefore, each term with exponent ≥1 is 10 mod 18. So:

Let me compute each term mod 18:

For the first term, 1*10^7 ≡ 1*10 mod 18 = 10

Second term: 1*10^6 ≡ 1*10 mod 18 = 10

Third term: 2*10^5 ≡ 2*10 = 20 mod 18 = 2

Fourth term: 1*10^4 ≡ 1*10 = 10

Fifth term: 3*10^3 ≡ 3*10 = 30 mod 18. 30 - 18 = 12, so 12

Sixth term: 1*10^2 ≡ 1*10 = 10

Seventh term: 4*10^1 ≡ 4*10 = 40 mod 18. 40 - 2*18 = 40 - 36 = 4

Eighth term: 1*10^0 = 1

Now, adding all these up:

Start with 10 (from first term)

+10 = 20

+2 = 22

+10 = 32

+12 = 44

+10 = 54

+4 = 58

+1 = 59

So total is 59 mod 18. Now compute 59 divided by 18. 18*3=54, so 59 - 54 = 5. Therefore, 59 ≡ 5 mod 18.

So that confirms that 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18. Then, -5 + 18 = 13, so -5 ≡ 13 mod 18. Therefore, n = 13.

But let me check once more with another approach. Maybe compute the number mod 18 by using the fact that 10 ≡ 10 mod 18, 10^2 ≡ 10 mod 18, etc., but maybe there's a pattern. Alternatively, since 10 ≡ -8 mod 18, maybe that helps? Let me try that.

If 10 ≡ -8 mod 18, then 10^1 ≡ -8, 10^2 ≡ (-8)*(-8) = 64 mod 18. 64 divided by 18 is 3*18=54, 64-54=10. So 10^2 ≡ 10 mod 18. Which matches the previous result. Then 10^3 ≡ 10*10 ≡ 10*10 = 100 ≡ 10 mod 18. So same as before. So regardless of the exponent, 10^k ≡ 10 mod 18 for k ≥ 1.

Therefore, the same calculation applies. So the total is 5 mod 18. Therefore, -5 mod 18 is 13.

Alternatively, maybe I can compute the original number mod 18 by breaking it down into smaller parts. For example, compute 11213141 divided by 18, but that would be tedious. Alternatively, use the fact that 18*622,952 = 11,213,136. Let me check that.

Wait, 18*600,000 = 10,800,000. Then 18*22,952 = ?

Wait, maybe this is too time-consuming. Alternatively, since we already have two methods giving 5 mod 18, which seems consistent, maybe I can trust that.

But just to be thorough, let me try another way. Let me compute 11213141 mod 18 by using the fact that 18*622,952 = 11,213,136. Then 11,213,141 - 11,213,136 = 5. Therefore, 11213141 ≡ 5 mod 18. Yes! So that confirms it. Therefore, -11213141 ≡ -5 mod 18, which is 13 mod 18.

Therefore, the integer n is 13.

But just to make sure, let me check with another method. Let me compute the number mod 18 by using the fact that 10 ≡ 10 mod 18, so we can compute the number as follows:

Starting from the rightmost digit, we can compute the value step by step. Let me try that.

The number is 11213141. Let me write it as digits: 1, 1, 2, 1, 3, 1, 4, 1. From left to right. Wait, but when computing mod 18, maybe it's easier to process from right to left? Let me think.

Alternatively, here's a method: take the number and repeatedly take mod 18 as you go. For example, starting with 0, then for each digit, compute (current_value * 10 + digit) mod 18.

Let me try that. Let's take the number 11213141. Let's process each digit from left to right:

Start with 0.

First digit: 1. Compute (0 * 10 + 1) mod 18 = 1 mod 18 = 1.

Second digit: 1. Compute (1 * 10 + 1) = 11 mod 18 = 11.

Third digit: 2. Compute (11 * 10 + 2) = 112 mod 18. Let me compute 112 / 18: 18*6=108, 112-108=4. So 4.

Fourth digit: 1. Compute (4 * 10 + 1) = 41 mod 18. 18*2=36, 41-36=5. So 5.

Fifth digit: 3. Compute (5 * 10 + 3) = 53 mod 18. 18*2=36, 53-36=17. 17 mod 18 is 17.

Sixth digit: 1. Compute (17 * 10 + 1) = 171 mod 18. 18*9=162, 171-162=9. So 9.

Seventh digit: 4. Compute (9 * 10 + 4) = 94 mod 18. 18*5=90, 94-90=4. So 4.

Eighth digit: 1. Compute (4 * 10 + 1) = 41 mod 18. As before, 41 - 36 = 5. So 5.

Wait, so according to this method, the result is 5 mod 18. Which matches our previous results. Therefore, 11213141 ≡ 5 mod 18. Therefore, -11213141 ≡ -5 mod 18. Which is 13 mod 18.

Therefore, n = 13.

But just to be absolutely sure, let me check with another approach. Let me compute 11213141 divided by 18 and find the remainder.

But doing this manually would be time-consuming, but maybe I can use the fact that 18*600,000 = 10,800,000. Then subtract that from 11,213,141:

11,213,141 - 10,800,000 = 413,141.

Now compute 413,141 divided by 18. Let me find how many times 18 goes into 413,141.

First, 18*20,000 = 360,000. Subtract that from 413,141: 413,141 - 360,000 = 53,141.

Now, 18*3,000 = 54,000. That's more than 53,141. So try 18*2,950 = 18*(3,000 - 50) = 54,000 - 900 = 53,100.

Subtract 53,100 from 53,141: 53,141 - 53,100 = 41.

So total so far is 600,000 + 20,000 + 2,950 = 622,950. And the remainder is 41.

Now, 41 divided by 18 is 2*18=36, remainder 5. Therefore, total remainder is 5. Therefore, 11,213,141 divided by 18 gives a remainder of 5. Therefore, 11213141 ≡ 5 mod 18.

Therefore, -11213141 ≡ -5 mod 18. Which is 13 mod 18.

Therefore, n = 13.

So all methods lead to the same conclusion. Therefore, I can be confident that the answer is 13.

But just to recap, the steps were:

1. Recognize that finding -11213141 mod 18 is equivalent to finding (-11213141) mod 18, which can be converted to a positive number by adding 18 until it's within 0-17.

2. Instead of dealing with the large number directly, use properties of modular arithmetic to simplify.

3. Noted that modulus 18 can be broken down into modulus 9 and modulus 2, then used Chinese Remainder Theorem to find the value mod 18.

4. Verified using multiple methods: digit sum for modulus 9, breaking down the number into powers of 10, processing digits step by step, and direct division to find the remainder.

All methods confirmed that 11213141 ≡ 5 mod 18, so -11213141 ≡ -5 ≡ 13 mod 18.

Therefore, the integer n is 13.

**Final Answer**
The integer $ n $ is \boxed{13}.

To solve the problem of finding an integer $ n $ such that:

$$
0 \leq n < 18 \quad \text{and} \quad n \equiv -11213141 \pmod{18}
$$

we aim to compute the value of $ -11213141 \mod 18 $, and then ensure the result lies in the range $ [0, 17] $.

---

### Step 1: Understand the Problem

We are given a large negative number, $ -11213141 $, and asked to find its residue modulo 18. Since the modulus is 18, the result must be an integer between 0 and 17 (inclusive). That is, we are looking for:

$$
n \equiv -11213141 \pmod{18}
$$

This is equivalent to:

$$
n \equiv (-11213141 \mod 18) \pmod{18}
$$

So, the key is to compute $ -11213141 \mod 18 $, and then adjust it to the range $ [0, 17] $ if necessary.

---

### Step 2: Compute $ 11213141 \mod 18 $

Rather than directly computing $ -11213141 \mod 18 $, we can first compute $ 11213141 \mod 18 $, and then take the negative of that result modulo 18.

#### Method: Use the fact that $ 10^k \equiv 10 \mod 18 $ for $ k \geq 1 $

We can break the number into its digits and use the fact that:

$$
10^k \equiv 10 \mod 18 \quad \text{for all } k \geq 1
$$

So, for the number $ 11213141 $, we can write it as:

$$
1 \cdot 10^7 + 1 \cdot 10^6 + 2 \cdot 10^5 + 1 \cdot 10^4 + 3 \cdot 10^3 + 1 \cdot 10^2 + 4 \cdot 10^1 + 1 \cdot 10^0
$$

Now, since $ 10^k \equiv 10 \mod 18 $ for $ k \geq 1 $, and $ 10^0 = 1 $, we can compute:

$$
\begin{align*}
1 \cdot 10 &\equiv 10 \\
1 \cdot 10 &\equiv 10 \\
2 \cdot 10 &\equiv 20 \equiv 2 \mod 18 \\
1 \cdot 10 &\equiv 10 \\
3 \cdot 10 &\equiv 30 \equiv 12 \mod 18 \\
1 \cdot 10 &\equiv 10 \\
4 \cdot 10 &\equiv 40 \equiv 4 \mod 18 \\
1 \cdot 1 &\equiv 1 \\
\end{align*}
$$

Now sum these values:

$$
10 + 10 + 2 + 10 + 12 + 10 + 4 + 1 = 59
$$

Now compute $ 59 \mod 18 $:

$$
59 \div 18 = 3 \text{ remainder } 5 \Rightarrow 59 \equiv 5 \mod 18
$$

So:

$$
11213141 \equiv 5 \mod 18
$$

---

### Step 3: Compute $ -11213141 \mod 18 $

Now that we know:

$$
11213141 \equiv 5 \mod 18
$$

Then:

$$
-11213141 \equiv -5 \mod 18
$$

To convert this to a positive value in the range $ [0, 17] $, we add 18:

$$
-5 + 18 = 13
$$

So:

$$
-11213141 \equiv 13 \mod 18
$$

---

### Final Answer

$$
\boxed{13}
$$

Solution: An integer is divisible by $18$ if and only if the sum of its digits is divisible by $9$ and the last digit is even (meaning it is divisible by both 9 and 2).  The sum of the digits of $-11213141$ is 14.  Since $-11213141$ is negative, this number is 5 $\textit{less than}$ a multiple of 9.  The number is 4 $\textit{more than}$ a multiple of 9.  Subtracting 4 gives  \[-11213141 = -11213145+4.\]Since $-11213145$ has a digit sum of 18, this number is a multiple of 9.  However this is not a multiple of 18 so we need to subtract 9 again: \[-11213141 = -11213154+13.\]Now the number $-11213154$ is a multiple of 18, so the answer is $\boxed{13}$. $$-11213141\equiv 13\pmod {18}.$$


Explanation:
Okay, so I need to find the value of f(3) given that f(-3) = 2 for the function f(x) = ax⁴ - bx² + x + 5. Hmm, let me think about how to approach this.

First, let me write down the function again to make sure I have it right: f(x) = a x⁴ - b x² + x + 5. So it's a quartic function with coefficients a and b. The problem gives me f(-3) = 2, and I need to find f(3).

Since the function has both even and odd powers of x, maybe there's some symmetry I can exploit here. Let me recall that for functions involving even and odd powers, sometimes substituting -x can lead to simplifications. Let me try that.

Let me compute f(-x). If I replace x with -x in the function, I get:

f(-x) = a(-x)⁴ - b(-x)² + (-x) + 5.

Calculating each term:

(-x)⁴ is x⁴ because any even power of a negative number is positive. Similarly, (-x)² is x². So substituting back:

f(-x) = a x⁴ - b x² - x + 5.

Now, let me compare this with the original function f(x) = a x⁴ - b x² + x + 5.

So, if I subtract f(-x) from f(x), let's see what happens:

f(x) - f(-x) = [a x⁴ - b x² + x + 5] - [a x⁴ - b x² - x + 5].

Let me compute term by term:

a x⁴ - a x⁴ = 0.

- b x² - (-b x²) = -b x² + b x² = 0.

Then, x - (-x) = x + x = 2x.

And 5 - 5 = 0.

So overall, f(x) - f(-x) = 2x.

That's interesting. So this tells me that f(x) - f(-x) = 2x.

Alternatively, maybe I can express f(x) + f(-x) as well? Let me check:

f(x) + f(-x) = [a x⁴ - b x² + x + 5] + [a x⁴ - b x² - x + 5] = 2a x⁴ - 2b x² + 10.

So that's another relation.

But how does this help me? Well, the problem gives me f(-3) = 2, and I need to find f(3). Let me denote f(3) as some value, say, y. Then, maybe I can set up equations using the relations I found.

Let me write down f(-3) and f(3) using the expressions above.

First, let's compute f(-3):

f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = a*(81) - b*(9) - 3 + 5 = 81a - 9b + 2.

And according to the problem, this equals 2. So:

81a - 9b + 2 = 2.

Subtracting 2 from both sides:

81a - 9b = 0.

Simplify this equation. Let me factor out 9:

9*(9a - b) = 0 => 9a - b = 0 => b = 9a.

Okay, so that's one equation relating a and b.

Now, let me compute f(3):

f(3) = a*(3)^4 - b*(3)^2 + 3 + 5 = 81a - 9b + 8.

Wait, so f(3) = 81a - 9b + 8.

But from earlier, we found that 81a - 9b = 0. Therefore, substituting that into f(3):

f(3) = 0 + 8 = 8.

Wait, so is it that straightforward? Let me check again.

Given that f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2 = 2. Therefore, 81a - 9b = 0. Then f(3) is 81a - 9b + 3 + 5 = 81a - 9b + 8. Since 81a - 9b is zero, f(3) is 0 + 8 = 8.

So the answer is 8? That seems too straightforward. Let me verify with another approach to make sure I didn't make a mistake.

Alternatively, using the relation f(x) - f(-x) = 2x. Let me plug in x = 3. Then:

f(3) - f(-3) = 2*3 = 6.

We know that f(-3) = 2, so:

f(3) - 2 = 6 => f(3) = 6 + 2 = 8.

Yes! That's the same result. So this confirms that f(3) is 8.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Let me check if there's another way to think about this. For example, considering the function's even and odd parts.

The function f(x) can be split into even and odd components. The even part would be the terms with even powers of x, and the odd part would be the terms with odd powers.

Looking at f(x) = ax⁴ - bx² + x + 5.

The even part is ax⁴ - bx² + 5, and the odd part is x.

So, f(x) = [ax⁴ - bx² + 5] + x.

Now, if we consider the even function E(x) = ax⁴ - bx² + 5 and the odd function O(x) = x.

Then, f(-x) = E(-x) + O(-x) = E(x) - O(x).

Therefore, f(-x) = E(x) - O(x).

So, f(x) + f(-x) = 2E(x), and f(x) - f(-x) = 2O(x).

Which is consistent with what I found earlier.

So, since O(x) = x, then f(x) - f(-x) = 2x.

Therefore, if we take x = 3, then f(3) - f(-3) = 6. Since f(-3) = 2, then f(3) = 6 + 2 = 8.

Same result again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Wait, but let me check if there's any possibility that the equations could have more constraints. For example, if I had more information, but here we only have one equation (from f(-3) = 2) and two variables (a and b). However, when calculating f(3), the terms involving a and b cancel out, so we don't need to know their exact values.

Let me verify with actual numbers. Suppose I choose a value for a, then compute b, and check f(3).

For example, let me pick a = 1. Then from b = 9a, we have b = 9.

So then f(x) = x⁴ - 9x² + x + 5.

Compute f(-3):

(-3)^4 - 9*(-3)^2 + (-3) + 5 = 81 - 9*9 - 3 + 5 = 81 - 81 - 3 + 5 = 0 - 3 + 5 = 2.

Which matches the given condition. Now compute f(3):

3^4 - 9*(3)^2 + 3 + 5 = 81 - 81 + 3 + 5 = 0 + 8 = 8.

Perfect, that works.

Let me try another value for a. Let me choose a = 0. Then b = 9*0 = 0.

So f(x) = 0x⁴ - 0x² + x + 5 = x + 5.

Then f(-3) = -3 + 5 = 2, which is correct. Then f(3) = 3 + 5 = 8.

Same result.

Another example: let me take a = 2, so b = 18.

Then f(x) = 2x⁴ - 18x² + x + 5.

Compute f(-3):

2*(81) - 18*(9) + (-3) + 5 = 162 - 162 - 3 + 5 = 0 - 3 + 5 = 2.

Good. Then f(3) = 2*(81) - 18*(9) + 3 + 5 = 162 - 162 + 3 + 5 = 0 + 8 = 8.

Same answer again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition from f(-3) = 2, f(3) is always 8.

Therefore, the answer is 8.

But just to make sure I didn't overlook anything, let me check if there's another way someone might approach this problem and perhaps make a mistake.

For example, someone might think that since f(-3) = 2, then f(3) is something else, maybe related to the coefficients. But since the function has both even and odd terms, the even terms will be the same for x and -x, while the odd terms will have opposite signs.

So, if I write f(3) = even_part(3) + odd_part(3), and f(-3) = even_part(-3) + odd_part(-3) = even_part(3) - odd_part(3).

Therefore, if I add f(3) and f(-3), I get 2*even_part(3). If I subtract f(-3) from f(3), I get 2*odd_part(3).

In this case, the odd_part(3) is just the coefficient of x times 3, which is 1*3 = 3. Therefore, f(3) - f(-3) = 2*3 = 6. Therefore, f(3) = f(-3) + 6 = 2 + 6 = 8.

This is another way to see it, using the even and odd parts.

So, all methods lead to the same conclusion. Therefore, I can be confident that the answer is 8.

I think that's thorough enough. I considered multiple approaches, checked with examples, and verified the relationships between f(x) and f(-x). All roads lead to Rome here.

**Final Answer**
The value of $ f(3) $ is \boxed{8}.

To determine the value of $ f(3) $, we are given the function:

$$
f(x) = ax^4 - bx^2 + x + 5
$$

and the condition:

$$
f(-3) = 2
$$

---

### Step 1: Use the given condition to find a relationship between $ a $ and $ b $

We substitute $ x = -3 $ into the function:

$$
f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = 81a - 9b - 3 + 5 = 81a - 9b + 2
$$

We are told that $ f(-3) = 2 $, so:

$$
81a - 9b + 2 = 2
$$

Subtracting 2 from both sides:

$$
81a - 9b = 0
$$

Divide both sides by 9:

$$
9a - b = 0 \quad \Rightarrow \quad b = 9a
$$

---

### Step 2: Use the relationship to find $ f(3) $

Now compute $ f(3) $:

$$
f(3) = a(3)^4 - b(3)^2 + 3 + 5 = 81a - 9b + 8
$$

Substitute $ b = 9a $:

$$
f(3) = 81a - 9(9a) + 8 = 81a - 81a + 8 = 8
$$

---

### Step 3: Confirm using symmetry

We can also observe that the function has both even and odd components:

- Even part: $ ax^4 - bx^2 + 5 $
- Odd part: $ x $

So, the function can be written as:

$$
f(x) = \text{even}(x) + \text{odd}(x)
$$

Then:

$$
f(-x) = \text{even}(x) - \text{odd}(x)
$$

So:

$$
f(x) - f(-x) = 2 \cdot \text{odd}(x) = 2x
$$

This gives us:

$$
f(3) - f(-3) = 2 \cdot 3 = 6
$$

Since $ f(-3) = 2 $, we find:

$$
f(3) = 2 + 6 = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of f(3) given that f(-3) = 2 for the function f(x) = ax⁴ - bx² + x + 5. Hmm, let me think about how to approach this.

First, let me write down the function again to make sure I have it right: f(x) = a x⁴ - b x² + x + 5. So it's a quartic function with coefficients a and b. The problem gives me f(-3) = 2, and I need to find f(3).

Since the function has both even and odd powers of x, maybe there's some symmetry I can exploit here. Let me recall that for functions involving even and odd powers, sometimes substituting -x can lead to simplifications. Let me try that.

Let me compute f(-x). If I replace x with -x in the function, I get:

f(-x) = a(-x)⁴ - b(-x)² + (-x) + 5.

Calculating each term:

(-x)⁴ is x⁴ because any even power of a negative number is positive. Similarly, (-x)² is x². So substituting back:

f(-x) = a x⁴ - b x² - x + 5.

Now, let me compare this with the original function f(x) = a x⁴ - b x² + x + 5.

So, if I subtract f(-x) from f(x), let's see what happens:

f(x) - f(-x) = [a x⁴ - b x² + x + 5] - [a x⁴ - b x² - x + 5].

Let me compute term by term:

a x⁴ - a x⁴ = 0.

- b x² - (-b x²) = -b x² + b x² = 0.

Then, x - (-x) = x + x = 2x.

And 5 - 5 = 0.

So overall, f(x) - f(-x) = 2x.

That's interesting. So this tells me that f(x) - f(-x) = 2x.

Alternatively, maybe I can express f(x) + f(-x) as well? Let me check:

f(x) + f(-x) = [a x⁴ - b x² + x + 5] + [a x⁴ - b x² - x + 5] = 2a x⁴ - 2b x² + 10.

So that's another relation.

But how does this help me? Well, the problem gives me f(-3) = 2, and I need to find f(3). Let me denote f(3) as some value, say, y. Then, maybe I can set up equations using the relations I found.

Let me write down f(-3) and f(3) using the expressions above.

First, let's compute f(-3):

f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = a*(81) - b*(9) - 3 + 5 = 81a - 9b + 2.

And according to the problem, this equals 2. So:

81a - 9b + 2 = 2.

Subtracting 2 from both sides:

81a - 9b = 0.

Simplify this equation. Let me factor out 9:

9*(9a - b) = 0 => 9a - b = 0 => b = 9a.

Okay, so that's one equation relating a and b.

Now, let me compute f(3):

f(3) = a*(3)^4 - b*(3)^2 + 3 + 5 = 81a - 9b + 8.

Wait, so f(3) = 81a - 9b + 8.

But from earlier, we found that 81a - 9b = 0. Therefore, substituting that into f(3):

f(3) = 0 + 8 = 8.

Wait, so is it that straightforward? Let me check again.

Given that f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2 = 2. Therefore, 81a - 9b = 0. Then f(3) is 81a - 9b + 3 + 5 = 81a - 9b + 8. Since 81a - 9b is zero, f(3) is 0 + 8 = 8.

So the answer is 8? That seems too straightforward. Let me verify with another approach to make sure I didn't make a mistake.

Alternatively, using the relation f(x) - f(-x) = 2x. Let me plug in x = 3. Then:

f(3) - f(-3) = 2*3 = 6.

We know that f(-3) = 2, so:

f(3) - 2 = 6 => f(3) = 6 + 2 = 8.

Yes! That's the same result. So this confirms that f(3) is 8.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Let me check if there's another way to think about this. For example, considering the function's even and odd parts.

The function f(x) can be split into even and odd components. The even part would be the terms with even powers of x, and the odd part would be the terms with odd powers.

Looking at f(x) = ax⁴ - bx² + x + 5.

The even part is ax⁴ - bx² + 5, and the odd part is x.

So, f(x) = [ax⁴ - bx² + 5] + x.

Now, if we consider the even function E(x) = ax⁴ - bx² + 5 and the odd function O(x) = x.

Then, f(-x) = E(-x) + O(-x) = E(x) - O(x).

Therefore, f(-x) = E(x) - O(x).

So, f(x) + f(-x) = 2E(x), and f(x) - f(-x) = 2O(x).

Which is consistent with what I found earlier.

So, since O(x) = x, then f(x) - f(-x) = 2x.

Therefore, if we take x = 3, then f(3) - f(-3) = 6. Since f(-3) = 2, then f(3) = 6 + 2 = 8.

Same result again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Wait, but let me check if there's any possibility that the equations could have more constraints. For example, if I had more information, but here we only have one equation (from f(-3) = 2) and two variables (a and b). However, when calculating f(3), the terms involving a and b cancel out, so we don't need to know their exact values.

Let me verify with actual numbers. Suppose I choose a value for a, then compute b, and check f(3).

For example, let me pick a = 1. Then from b = 9a, we have b = 9.

So then f(x) = x⁴ - 9x² + x + 5.

Compute f(-3):

(-3)^4 - 9*(-3)^2 + (-3) + 5 = 81 - 9*9 - 3 + 5 = 81 - 81 - 3 + 5 = 0 - 3 + 5 = 2.

Which matches the given condition. Now compute f(3):

3^4 - 9*(3)^2 + 3 + 5 = 81 - 81 + 3 + 5 = 0 + 8 = 8.

Perfect, that works.

Let me try another value for a. Let me choose a = 0. Then b = 9*0 = 0.

So f(x) = 0x⁴ - 0x² + x + 5 = x + 5.

Then f(-3) = -3 + 5 = 2, which is correct. Then f(3) = 3 + 5 = 8.

Same result.

Another example: let me take a = 2, so b = 18.

Then f(x) = 2x⁴ - 18x² + x + 5.

Compute f(-3):

2*(81) - 18*(9) + (-3) + 5 = 162 - 162 - 3 + 5 = 0 - 3 + 5 = 2.

Good. Then f(3) = 2*(81) - 18*(9) + 3 + 5 = 162 - 162 + 3 + 5 = 0 + 8 = 8.

Same answer again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition from f(-3) = 2, f(3) is always 8.

Therefore, the answer is 8.

But just to make sure I didn't overlook anything, let me check if there's another way someone might approach this problem and perhaps make a mistake.

For example, someone might think that since f(-3) = 2, then f(3) is something else, maybe related to the coefficients. But since the function has both even and odd terms, the even terms will be the same for x and -x, while the odd terms will have opposite signs.

So, if I write f(3) = even_part(3) + odd_part(3), and f(-3) = even_part(-3) + odd_part(-3) = even_part(3) - odd_part(3).

Therefore, if I add f(3) and f(-3), I get 2*even_part(3). If I subtract f(-3) from f(3), I get 2*odd_part(3).

In this case, the odd_part(3) is just the coefficient of x times 3, which is 1*3 = 3. Therefore, f(3) - f(-3) = 2*3 = 6. Therefore, f(3) = f(-3) + 6 = 2 + 6 = 8.

This is another way to see it, using the even and odd parts.

So, all methods lead to the same conclusion. Therefore, I can be confident that the answer is 8.

I think that's thorough enough. I considered multiple approaches, checked with examples, and verified the relationships between f(x) and f(-x). All roads lead to Rome here.

**Final Answer**
The value of $ f(3) $ is \boxed{8}.

To determine the value of $ f(3) $, we are given the function:

$$
f(x) = ax^4 - bx^2 + x + 5
$$

and the condition:

$$
f(-3) = 2
$$

---

### Step 1: Use the given condition to find a relationship between $ a $ and $ b $

We substitute $ x = -3 $ into the function:

$$
f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = 81a - 9b - 3 + 5 = 81a - 9b + 2
$$

We are told that $ f(-3) = 2 $, so:

$$
81a - 9b + 2 = 2
$$

Subtracting 2 from both sides:

$$
81a - 9b = 0
$$

Divide both sides by 9:

$$
9a - b = 0 \quad \Rightarrow \quad b = 9a
$$

---

### Step 2: Use the relationship to find $ f(3) $

Now compute $ f(3) $:

$$
f(3) = a(3)^4 - b(3)^2 + 3 + 5 = 81a - 9b + 8
$$

Substitute $ b = 9a $:

$$
f(3) = 81a - 9(9a) + 8 = 81a - 81a + 8 = 8
$$

---

### Step 3: Confirm using symmetry

We can also observe that the function has both even and odd components:

- Even part: $ ax^4 - bx^2 + 5 $
- Odd part: $ x $

So, the function can be written as:

$$
f(x) = \text{even}(x) + \text{odd}(x)
$$

Then:

$$
f(-x) = \text{even}(x) - \text{odd}(x)
$$

So:

$$
f(x) - f(-x) = 2 \cdot \text{odd}(x) = 2x
$$

This gives us:

$$
f(3) - f(-3) = 2 \cdot 3 = 6
$$

Since $ f(-3) = 2 $, we find:

$$
f(3) = 2 + 6 = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

Solution: Evaluating $f(x)$ for $x=3$ and $x=-3$, we have \[\left\{ \begin{aligned} f(3)& = a \cdot 3^4 - b \cdot 3^2 + 3 + 5, \\ f(-3) &= a \cdot (-3)^4 - b \cdot (-3)^2 + (-3) + 5. \end{aligned} \right.\]If we subtract the second equation from the first equation, all the terms but one cancel out, and we get \[f(3) - f(-3) = 3 - (-3) = 6.\]Thus, if $f(-3) = 2,$ then $f(3) = f(-3) + 6 = 2 + 6 = \boxed{8}.$


Explanation:
Okay, so I need to find the equation of a hyperbola given two foci, F₁ = (10, 2) and F₂ = (-16, 2), and the condition that the absolute difference of the distances from any point P on the hyperbola to the two foci is 24. Then, once I have the equation in the standard form, I need to find h + k + a + b. Alright, let me recall what I know about hyperbolas.

First, a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. That constant is usually denoted as 2a, where a is the distance from the center to a vertex. So in this case, the given condition |PF₁ - PF₂| = 24 implies that 2a = 24, so a = 12. Got that.

Next, I need to find the center of the hyperbola. The center is the midpoint between the two foci. Let me calculate that. The midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2). Plugging in the coordinates of F₁ and F₂:

Midpoint h, k = ((10 + (-16))/2, (2 + 2)/2) = ((-6)/2, 4/2) = (-3, 2). So the center is at (-3, 2). Therefore, h = -3 and k = 2. So far, so good.

Now, I need to figure out the values of a, b, and the orientation of the hyperbola. Wait, I already found a = 12. But I need to find b. For hyperbolas, the relationship between a, b, and the distance between the foci (which is 2c) is given by c² = a² + b². So I need to find c first.

The distance between the two foci is the distance between F₁ and F₂. Since both foci have the same y-coordinate, 2, the distance between them is just the difference in the x-coordinates. Let me compute that:

Distance between F₁ and F₂ = |10 - (-16)| = |10 + 16| = 26. Therefore, 2c = 26, so c = 13.

Now, using the relationship c² = a² + b², I can solve for b. Let me plug in the values:

c² = 13² = 169

a² = 12² = 144

So, 169 = 144 + b² => b² = 169 - 144 = 25 => b = 5.

Wait, so b is 5? Let me check that again. Yes, 13 squared is 169, 12 squared is 144, subtracting gives 25, square root is 5. So b = 5.

Now, I need to determine the orientation of the hyperbola. Since the foci are located at (-16, 2) and (10, 2), they lie on the horizontal line y = 2. Therefore, the transverse axis is horizontal. That means the hyperbola opens left and right. Therefore, the standard form of the equation is:

[(x - h)²]/a² - [(y - k)²]/b² = 1

Which, with h = -3, k = 2, a² = 144, b² = 25, becomes:

[(x - (-3))²]/144 - [(y - 2)²]/25 = 1

Simplifying the terms:

[(x + 3)²]/144 - [(y - 2)²]/25 = 1

So, the equation is as above. Therefore, h = -3, k = 2, a = 12, b = 5. The question asks for h + k + a + b. Let me compute that:

h + k + a + b = (-3) + 2 + 12 + 5 = (-3 + 2) + (12 + 5) = (-1) + 17 = 16.

Wait, that seems straightforward, but let me double-check all steps to make sure I didn't make a mistake.

First, midpoint calculation: (10 + (-16))/2 = (-6)/2 = -3, and (2 + 2)/2 = 2. Correct. So center at (-3, 2). Then, distance between foci: 10 - (-16) is 26, so 2c = 26, c = 13. Then, since 2a = 24, a = 12. Then c² = a² + b² => 169 = 144 + b² => b² = 25 => b = 5. Correct. Then, since the foci are on horizontal line, hyperbola is horizontal, so standard form is [(x - h)^2]/a² - [(y - k)^2]/b² = 1. So equation is [(x + 3)^2]/144 - [(y - 2)^2]/25 = 1. Therefore, h = -3, k = 2, a = 12, b = 5. Adding them up: -3 + 2 + 12 + 5. Let me compute again:

-3 + 2 is -1, 12 + 5 is 17, so total is -1 + 17 = 16. So the answer is 16? Wait, but let me check if the hyperbola is indeed horizontal. Since the foci are on the horizontal line, yes, the transverse axis is horizontal, so the hyperbola opens left and right, so the standard form is correct.

Alternatively, if the foci were on a vertical line, the equation would have y terms first. But here, since they are on horizontal line, x terms first. So that's correct.

Another thing to check: the value of a. The definition says |PF₁ - PF₂| = 2a. So 2a = 24, so a = 12. Correct.

Also, c is the distance from center to each focus. The center is at (-3, 2). Let me check the distance from center to F₁. F₁ is at (10, 2). The distance between (-3, 2) and (10, 2) is |10 - (-3)| = 13. Which is c = 13. Correct. Similarly, distance from center to F₂ is |-16 - (-3)| = |-13| = 13. Correct. So c is indeed 13.

Therefore, all steps check out. Therefore, h + k + a + b = -3 + 2 + 12 + 5 = 16.

But wait, let me check once more the arithmetic:

-3 + 2 = -1

12 + 5 = 17

-1 + 17 = 16. Yes.

So the answer is 16. Therefore, I think that's correct.

But just to be thorough, let me try to write the equation of the hyperbola and verify with a sample point. For example, let's take a point on the hyperbola. Since the center is at (-3, 2), and a = 12, the vertices are at (-3 ± 12, 2). So vertices at (9, 2) and (-15, 2). Let me check the difference in distances from one of these vertices to the foci. Let's take (9, 2).

Distance to F₁ (10, 2): sqrt[(9 - 10)^2 + (2 - 2)^2] = sqrt[1 + 0] = 1.

Distance to F₂ (-16, 2): sqrt[(9 - (-16))^2 + (2 - 2)^2] = sqrt[(25)^2] = 25.

So |1 - 25| = 24. Which matches the given condition. Similarly, check the other vertex (-15, 2):

Distance to F₁ (10, 2): sqrt[(-15 - 10)^2 + 0] = sqrt[(-25)^2] = 25.

Distance to F₂ (-16, 2): sqrt[(-15 - (-16))^2 + 0] = sqrt[1^2] = 1.

So |25 - 1| = 24. Correct. So the vertices satisfy the condition.

Another check: let me take a point on the hyperbola. For example, suppose we take a point along the x-axis. Wait, the center is at (-3, 2). Let me think. Maybe take a point that's a = 12 to the right of the center. So center is (-3, 2), so adding 12 to x-coordinate: (-3 + 12, 2) = (9, 2), which is the vertex we already checked. Similarly, left side is (-3 - 12, 2) = (-15, 2).

Alternatively, let me take a point not on the axis. For example, suppose we take a point (x, y) such that it's on the hyperbola. Let me try to find another point. For example, suppose y = 2, so the horizontal line through the center. Then, the hyperbola equation becomes [(x + 3)^2]/144 - 0 = 1 => (x + 3)^2 = 144 => x + 3 = ±12 => x = 9 or x = -15. Which are the vertices. So that's consistent.

Alternatively, take a point above the center. Let me try to find a point. Let me suppose that the hyperbola has a point (x, y) where y ≠ 2. Let me try to find such a point. For example, suppose we take a point where the y-coordinate is 2 + b, which is 2 + 5 = 7. Let me see if there's a corresponding x.

But maybe that's too arbitrary. Alternatively, use the definition. Let me take a point P such that PF₁ - PF₂ = 24. Let me try to find such a point. For example, take a point on the right branch. Let me suppose that P is on the right branch, so PF₁ - PF₂ = 24. Let me take a point that's far to the right. Let me suppose that P is (x, 2), so on the horizontal line. Then, PF₁ = |x - 10|, PF₂ = |x - (-16)| = |x + 16|. Then, PF₁ - PF₂ = |x - 10| - |x + 16|. For x > 10, this is (x - 10) - (x + 16) = -26. But absolute value is 26. Wait, but the condition is |PF₁ - PF₂| = 24. So maybe not on the axis.

Alternatively, maybe take a point where PF₁ - PF₂ = 24. Let me suppose that P is on the right branch. Let me take a point where PF₁ - PF₂ = 24. Let me suppose that P is (x, y). Then, sqrt[(x - 10)^2 + (y - 2)^2] - sqrt[(x + 16)^2 + (y - 2)^2] = 24. This seems complicated to solve, but maybe if I take a point that's on the hyperbola. For example, take the vertex (9, 2). As we saw, that gives 1 - 25 = -24, absolute value 24. So that works.

Alternatively, take a point that's not on the axis. Let me try to find another point. Let me think. Suppose we take a point where y = 2 + b = 7. Let me try to find x such that [(x + 3)^2]/144 - [(7 - 2)^2]/25 = 1. So [(x + 3)^2]/144 - [25]/25 = 1 => [(x + 3)^2]/144 - 1 = 1 => [(x + 3)^2]/144 = 2 => (x + 3)^2 = 288 => x + 3 = ±√288 = ±12√2. Therefore, x = -3 ± 12√2. So the points (-3 + 12√2, 7) and (-3 - 12√2, 7) are on the hyperbola. Let me check one of them. Let's take (-3 + 12√2, 7). Let me compute PF₁ and PF₂.

First, compute PF₁: distance from (-3 + 12√2, 7) to (10, 2).

Difference in x: (-3 + 12√2 - 10) = -13 + 12√2

Difference in y: 7 - 2 = 5

So PF₁ = sqrt[(-13 + 12√2)^2 + 5^2]

Similarly, PF₂: distance to (-16, 2):

Difference in x: (-3 + 12√2 - (-16)) = 13 + 12√2

Difference in y: same 5

So PF₂ = sqrt[(13 + 12√2)^2 + 5^2]

Now, compute |PF₁ - PF₂|. Let me compute PF₁ - PF₂. Let me denote A = -13 + 12√2 and B = 13 + 12√2. Then PF₁ = sqrt(A² + 25), PF₂ = sqrt(B² + 25).

But since B = -A + 26? Wait, let me check:

A = -13 + 12√2

B = 13 + 12√2

So B = -A + 26? Let me compute -A + 26:

-(-13 + 12√2) + 26 = 13 - 12√2 + 26 = 39 - 12√2. Which is not equal to B. Wait, maybe not helpful.

Alternatively, note that B = 13 + 12√2, and A = -13 + 12√2. So B - A = 26.

But how does that help? Maybe not. Alternatively, since the point is on the hyperbola, we know that |PF₁ - PF₂| = 24. So maybe I can trust the equation. Since we derived the equation from the definition, and the vertex checks out, maybe this is sufficient.

Alternatively, maybe I can compute PF₁ - PF₂. Let me try to approximate numerically. Let me compute approximate values. Let me take √2 ≈ 1.4142. Then 12√2 ≈ 16.97.

So A = -13 + 16.97 ≈ 3.97

B = 13 + 16.97 ≈ 29.97

So PF₁ ≈ sqrt( (3.97)^2 + 5^2 ) ≈ sqrt(15.76 + 25) ≈ sqrt(40.76) ≈ 6.38

PF₂ ≈ sqrt( (29.97)^2 + 5^2 ) ≈ sqrt(898.2 + 25) ≈ sqrt(923.2) ≈ 30.38

Then PF₁ - PF₂ ≈ 6.38 - 30.38 ≈ -24. So absolute value is 24. Which matches the condition. So that works.

Therefore, the equation seems correct. Therefore, my calculation of h, k, a, b is correct, and their sum is 16.

Therefore, I think the answer is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given two foci of a hyperbola:
$$
F_1 = (10, 2), \quad F_2 = (-16, 2)
$$
and the condition that for any point $ P $ on the hyperbola, the absolute difference of distances to the foci is constant:
$$
|PF_1 - PF_2| = 24
$$
This is the defining property of a **hyperbola**, where the constant difference is $ 2a $, so we immediately deduce:
$$
2a = 24 \Rightarrow a = 12
$$

---

### Step 1: Find the Center of the Hyperbola

The center of the hyperbola is the **midpoint** of the segment joining the two foci. Using the midpoint formula:
$$
\text{Center} = \left( \frac{10 + (-16)}{2}, \frac{2 + 2}{2} \right) = (-3, 2)
$$
So, the center is at $ (h, k) = (-3, 2) $

---

### Step 2: Determine the Value of $ c $

The distance between the foci is:
$$
\text{Distance} = |10 - (-16)| = 26 \Rightarrow 2c = 26 \Rightarrow c = 13
$$

---

### Step 3: Use the Hyperbola Identity to Find $ b $

For a hyperbola, the relationship between $ a $, $ b $, and $ c $ is:
$$
c^2 = a^2 + b^2
$$
Substituting the known values:
$$
13^2 = 12^2 + b^2 \Rightarrow 169 = 144 + b^2 \Rightarrow b^2 = 25 \Rightarrow b = 5
$$

---

### Step 4: Determine the Orientation of the Hyperbola

Since the foci lie on the same horizontal line $ y = 2 $, the transverse axis is **horizontal**, and the hyperbola opens left and right. Therefore, the standard form of the equation is:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$
Substituting the known values:
$$
\frac{(x + 3)^2}{144} - \frac{(y - 2)^2}{25} = 1
$$

---

### Step 5: Compute the Final Answer

We are asked to find:
$$
h + k + a + b
$$
Substituting the values:
$$
h = -3, \quad k = 2, \quad a = 12, \quad b = 5
$$
$$
h + k + a + b = -3 + 2 + 12 + 5 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of a hyperbola given two foci, F₁ = (10, 2) and F₂ = (-16, 2), and the condition that the absolute difference of the distances from any point P on the hyperbola to the two foci is 24. Then, once I have the equation in the standard form, I need to find h + k + a + b. Alright, let me recall what I know about hyperbolas.

First, a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. That constant is usually denoted as 2a, where a is the distance from the center to a vertex. So in this case, the given condition |PF₁ - PF₂| = 24 implies that 2a = 24, so a = 12. Got that.

Next, I need to find the center of the hyperbola. The center is the midpoint between the two foci. Let me calculate that. The midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2). Plugging in the coordinates of F₁ and F₂:

Midpoint h, k = ((10 + (-16))/2, (2 + 2)/2) = ((-6)/2, 4/2) = (-3, 2). So the center is at (-3, 2). Therefore, h = -3 and k = 2. So far, so good.

Now, I need to figure out the values of a, b, and the orientation of the hyperbola. Wait, I already found a = 12. But I need to find b. For hyperbolas, the relationship between a, b, and the distance between the foci (which is 2c) is given by c² = a² + b². So I need to find c first.

The distance between the two foci is the distance between F₁ and F₂. Since both foci have the same y-coordinate, 2, the distance between them is just the difference in the x-coordinates. Let me compute that:

Distance between F₁ and F₂ = |10 - (-16)| = |10 + 16| = 26. Therefore, 2c = 26, so c = 13.

Now, using the relationship c² = a² + b², I can solve for b. Let me plug in the values:

c² = 13² = 169

a² = 12² = 144

So, 169 = 144 + b² => b² = 169 - 144 = 25 => b = 5.

Wait, so b is 5? Let me check that again. Yes, 13 squared is 169, 12 squared is 144, subtracting gives 25, square root is 5. So b = 5.

Now, I need to determine the orientation of the hyperbola. Since the foci are located at (-16, 2) and (10, 2), they lie on the horizontal line y = 2. Therefore, the transverse axis is horizontal. That means the hyperbola opens left and right. Therefore, the standard form of the equation is:

[(x - h)²]/a² - [(y - k)²]/b² = 1

Which, with h = -3, k = 2, a² = 144, b² = 25, becomes:

[(x - (-3))²]/144 - [(y - 2)²]/25 = 1

Simplifying the terms:

[(x + 3)²]/144 - [(y - 2)²]/25 = 1

So, the equation is as above. Therefore, h = -3, k = 2, a = 12, b = 5. The question asks for h + k + a + b. Let me compute that:

h + k + a + b = (-3) + 2 + 12 + 5 = (-3 + 2) + (12 + 5) = (-1) + 17 = 16.

Wait, that seems straightforward, but let me double-check all steps to make sure I didn't make a mistake.

First, midpoint calculation: (10 + (-16))/2 = (-6)/2 = -3, and (2 + 2)/2 = 2. Correct. So center at (-3, 2). Then, distance between foci: 10 - (-16) is 26, so 2c = 26, c = 13. Then, since 2a = 24, a = 12. Then c² = a² + b² => 169 = 144 + b² => b² = 25 => b = 5. Correct. Then, since the foci are on horizontal line, hyperbola is horizontal, so standard form is [(x - h)^2]/a² - [(y - k)^2]/b² = 1. So equation is [(x + 3)^2]/144 - [(y - 2)^2]/25 = 1. Therefore, h = -3, k = 2, a = 12, b = 5. Adding them up: -3 + 2 + 12 + 5. Let me compute again:

-3 + 2 is -1, 12 + 5 is 17, so total is -1 + 17 = 16. So the answer is 16? Wait, but let me check if the hyperbola is indeed horizontal. Since the foci are on the horizontal line, yes, the transverse axis is horizontal, so the hyperbola opens left and right, so the standard form is correct.

Alternatively, if the foci were on a vertical line, the equation would have y terms first. But here, since they are on horizontal line, x terms first. So that's correct.

Another thing to check: the value of a. The definition says |PF₁ - PF₂| = 2a. So 2a = 24, so a = 12. Correct.

Also, c is the distance from center to each focus. The center is at (-3, 2). Let me check the distance from center to F₁. F₁ is at (10, 2). The distance between (-3, 2) and (10, 2) is |10 - (-3)| = 13. Which is c = 13. Correct. Similarly, distance from center to F₂ is |-16 - (-3)| = |-13| = 13. Correct. So c is indeed 13.

Therefore, all steps check out. Therefore, h + k + a + b = -3 + 2 + 12 + 5 = 16.

But wait, let me check once more the arithmetic:

-3 + 2 = -1

12 + 5 = 17

-1 + 17 = 16. Yes.

So the answer is 16. Therefore, I think that's correct.

But just to be thorough, let me try to write the equation of the hyperbola and verify with a sample point. For example, let's take a point on the hyperbola. Since the center is at (-3, 2), and a = 12, the vertices are at (-3 ± 12, 2). So vertices at (9, 2) and (-15, 2). Let me check the difference in distances from one of these vertices to the foci. Let's take (9, 2).

Distance to F₁ (10, 2): sqrt[(9 - 10)^2 + (2 - 2)^2] = sqrt[1 + 0] = 1.

Distance to F₂ (-16, 2): sqrt[(9 - (-16))^2 + (2 - 2)^2] = sqrt[(25)^2] = 25.

So |1 - 25| = 24. Which matches the given condition. Similarly, check the other vertex (-15, 2):

Distance to F₁ (10, 2): sqrt[(-15 - 10)^2 + 0] = sqrt[(-25)^2] = 25.

Distance to F₂ (-16, 2): sqrt[(-15 - (-16))^2 + 0] = sqrt[1^2] = 1.

So |25 - 1| = 24. Correct. So the vertices satisfy the condition.

Another check: let me take a point on the hyperbola. For example, suppose we take a point along the x-axis. Wait, the center is at (-3, 2). Let me think. Maybe take a point that's a = 12 to the right of the center. So center is (-3, 2), so adding 12 to x-coordinate: (-3 + 12, 2) = (9, 2), which is the vertex we already checked. Similarly, left side is (-3 - 12, 2) = (-15, 2).

Alternatively, let me take a point not on the axis. For example, suppose we take a point (x, y) such that it's on the hyperbola. Let me try to find another point. For example, suppose y = 2, so the horizontal line through the center. Then, the hyperbola equation becomes [(x + 3)^2]/144 - 0 = 1 => (x + 3)^2 = 144 => x + 3 = ±12 => x = 9 or x = -15. Which are the vertices. So that's consistent.

Alternatively, take a point above the center. Let me try to find a point. Let me suppose that the hyperbola has a point (x, y) where y ≠ 2. Let me try to find such a point. For example, suppose we take a point where the y-coordinate is 2 + b, which is 2 + 5 = 7. Let me see if there's a corresponding x.

But maybe that's too arbitrary. Alternatively, use the definition. Let me take a point P such that PF₁ - PF₂ = 24. Let me try to find such a point. For example, take a point on the right branch. Let me suppose that P is on the right branch, so PF₁ - PF₂ = 24. Let me take a point that's far to the right. Let me suppose that P is (x, 2), so on the horizontal line. Then, PF₁ = |x - 10|, PF₂ = |x - (-16)| = |x + 16|. Then, PF₁ - PF₂ = |x - 10| - |x + 16|. For x > 10, this is (x - 10) - (x + 16) = -26. But absolute value is 26. Wait, but the condition is |PF₁ - PF₂| = 24. So maybe not on the axis.

Alternatively, maybe take a point where PF₁ - PF₂ = 24. Let me suppose that P is on the right branch. Let me take a point where PF₁ - PF₂ = 24. Let me suppose that P is (x, y). Then, sqrt[(x - 10)^2 + (y - 2)^2] - sqrt[(x + 16)^2 + (y - 2)^2] = 24. This seems complicated to solve, but maybe if I take a point that's on the hyperbola. For example, take the vertex (9, 2). As we saw, that gives 1 - 25 = -24, absolute value 24. So that works.

Alternatively, take a point that's not on the axis. Let me try to find another point. Let me think. Suppose we take a point where y = 2 + b = 7. Let me try to find x such that [(x + 3)^2]/144 - [(7 - 2)^2]/25 = 1. So [(x + 3)^2]/144 - [25]/25 = 1 => [(x + 3)^2]/144 - 1 = 1 => [(x + 3)^2]/144 = 2 => (x + 3)^2 = 288 => x + 3 = ±√288 = ±12√2. Therefore, x = -3 ± 12√2. So the points (-3 + 12√2, 7) and (-3 - 12√2, 7) are on the hyperbola. Let me check one of them. Let's take (-3 + 12√2, 7). Let me compute PF₁ and PF₂.

First, compute PF₁: distance from (-3 + 12√2, 7) to (10, 2).

Difference in x: (-3 + 12√2 - 10) = -13 + 12√2

Difference in y: 7 - 2 = 5

So PF₁ = sqrt[(-13 + 12√2)^2 + 5^2]

Similarly, PF₂: distance to (-16, 2):

Difference in x: (-3 + 12√2 - (-16)) = 13 + 12√2

Difference in y: same 5

So PF₂ = sqrt[(13 + 12√2)^2 + 5^2]

Now, compute |PF₁ - PF₂|. Let me compute PF₁ - PF₂. Let me denote A = -13 + 12√2 and B = 13 + 12√2. Then PF₁ = sqrt(A² + 25), PF₂ = sqrt(B² + 25).

But since B = -A + 26? Wait, let me check:

A = -13 + 12√2

B = 13 + 12√2

So B = -A + 26? Let me compute -A + 26:

-(-13 + 12√2) + 26 = 13 - 12√2 + 26 = 39 - 12√2. Which is not equal to B. Wait, maybe not helpful.

Alternatively, note that B = 13 + 12√2, and A = -13 + 12√2. So B - A = 26.

But how does that help? Maybe not. Alternatively, since the point is on the hyperbola, we know that |PF₁ - PF₂| = 24. So maybe I can trust the equation. Since we derived the equation from the definition, and the vertex checks out, maybe this is sufficient.

Alternatively, maybe I can compute PF₁ - PF₂. Let me try to approximate numerically. Let me compute approximate values. Let me take √2 ≈ 1.4142. Then 12√2 ≈ 16.97.

So A = -13 + 16.97 ≈ 3.97

B = 13 + 16.97 ≈ 29.97

So PF₁ ≈ sqrt( (3.97)^2 + 5^2 ) ≈ sqrt(15.76 + 25) ≈ sqrt(40.76) ≈ 6.38

PF₂ ≈ sqrt( (29.97)^2 + 5^2 ) ≈ sqrt(898.2 + 25) ≈ sqrt(923.2) ≈ 30.38

Then PF₁ - PF₂ ≈ 6.38 - 30.38 ≈ -24. So absolute value is 24. Which matches the condition. So that works.

Therefore, the equation seems correct. Therefore, my calculation of h, k, a, b is correct, and their sum is 16.

Therefore, I think the answer is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given two foci of a hyperbola:
$$
F_1 = (10, 2), \quad F_2 = (-16, 2)
$$
and the condition that for any point $ P $ on the hyperbola, the absolute difference of distances to the foci is constant:
$$
|PF_1 - PF_2| = 24
$$
This is the defining property of a **hyperbola**, where the constant difference is $ 2a $, so we immediately deduce:
$$
2a = 24 \Rightarrow a = 12
$$

---

### Step 1: Find the Center of the Hyperbola

The center of the hyperbola is the **midpoint** of the segment joining the two foci. Using the midpoint formula:
$$
\text{Center} = \left( \frac{10 + (-16)}{2}, \frac{2 + 2}{2} \right) = (-3, 2)
$$
So, the center is at $ (h, k) = (-3, 2) $

---

### Step 2: Determine the Value of $ c $

The distance between the foci is:
$$
\text{Distance} = |10 - (-16)| = 26 \Rightarrow 2c = 26 \Rightarrow c = 13
$$

---

### Step 3: Use the Hyperbola Identity to Find $ b $

For a hyperbola, the relationship between $ a $, $ b $, and $ c $ is:
$$
c^2 = a^2 + b^2
$$
Substituting the known values:
$$
13^2 = 12^2 + b^2 \Rightarrow 169 = 144 + b^2 \Rightarrow b^2 = 25 \Rightarrow b = 5
$$

---

### Step 4: Determine the Orientation of the Hyperbola

Since the foci lie on the same horizontal line $ y = 2 $, the transverse axis is **horizontal**, and the hyperbola opens left and right. Therefore, the standard form of the equation is:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$
Substituting the known values:
$$
\frac{(x + 3)^2}{144} - \frac{(y - 2)^2}{25} = 1
$$

---

### Step 5: Compute the Final Answer

We are asked to find:
$$
h + k + a + b
$$
Substituting the values:
$$
h = -3, \quad k = 2, \quad a = 12, \quad b = 5
$$
$$
h + k + a + b = -3 + 2 + 12 + 5 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

Solution:
The center of the hyperbola is the midpoint of $\overline{F_1 F_2},$ which is $(-3,2).$  Thus, $h = -3$ and $k = 2.$

Also, $2a = 24,$ so $a = 12.$  The distance between the foci is $2c = 26,$ so $c = 13.$  Then $b^2 = c^2 - a^2 = 169 - 144 = 25,$ so $b = 5.$

Hence, $h + k + a + b = (-3) + 2 + 12 + 5 = \boxed{16}.$


Explanation:
Okay, so I need to divide 413 base 5 by 2 base 5 and give the answer in base 5. Hmm, let me think. First, maybe I should convert both numbers to base 10, do the division there, and then convert the result back to base 5? That might be easier for me since I'm more comfortable with base 10 arithmetic. Let me try that approach.

Starting with the dividend, which is 413 in base 5. To convert that to base 10, I need to remember that each digit represents a power of 5. The rightmost digit is 5^0, then 5^1, then 5^2, and so on. So let's break down 413_5:

The digits are 4, 1, 3 from left to right. So that would be:

4 * 5^2 + 1 * 5^1 + 3 * 5^0.

Calculating each term:

First term: 4 * 5^2. 5 squared is 25, so 4 * 25 = 100.

Second term: 1 * 5^1 = 1 * 5 = 5.

Third term: 3 * 5^0 = 3 * 1 = 3.

Adding them up: 100 + 5 + 3 = 108. So 413 base 5 is 108 in base 10.

Now the divisor is 2_5. Converting that to base 10 is straightforward: 2 * 5^0 = 2 * 1 = 2. So 2 base 5 is 2 in base 10.

So now the problem becomes 108 divided by 2 in base 10. Let me do that division. 108 divided by 2 is 54. So the result in base 10 is 54. Now I need to convert 54 back to base 5.

To convert 54 to base 5, I need to find the coefficients for the powers of 5. Let me recall that 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125. Since 5^3 is 125 which is larger than 54, the highest power needed is 5^2 = 25.

So let's divide 54 by 25 to find the coefficient for 5^2. 54 divided by 25 is 2 with a remainder. 2*25 = 50. So the coefficient for 5^2 is 2. Then subtract 50 from 54, which leaves 4.

Now, moving to the next lower power, which is 5^1 = 5. Take the remainder 4 and divide by 5. That's 0 with a remainder of 4. So the coefficient for 5^1 is 0.

Then the remainder is 4, which is the coefficient for 5^0. So putting it all together, the base 5 number is 2 (for 5^2), 0 (for 5^1), and 4 (for 5^0). Therefore, 54 in base 10 is 204 in base 5.

Wait, let me check that again. Let me verify the conversion of 204 base 5 back to base 10 to make sure. 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Yes, that's correct. So 54 base 10 is indeed 204 base 5. Therefore, the quotient of 413_5 divided by 2_5 is 204_5.

But hold on, maybe I should try doing the division directly in base 5 to confirm? Sometimes converting back and forth can introduce errors, especially if I made a mistake in conversion. Let me try that method as well to verify.

So, let's perform the division 413_5 ÷ 2_5 directly in base 5. Let me recall how division works in different bases. It's similar to base 10, but all the operations are done in base 5.

First, let me write down the dividend and divisor:

Dividend: 413_5

Divisor: 2_5

So, we need to divide 413_5 by 2_5. Let me set it up like long division.

First, I need to see how many times 2_5 goes into the first digit of the dividend. The first digit is 4_5. Let me convert 4_5 to base 10 to understand its value. 4 in base 5 is 4 in base 10. So 4 divided by 2 is 2. So 2_5 goes into 4_5 twice. So I write 2 as the first digit of the quotient.

Multiply 2_5 by 2_5. Let me compute that in base 5. 2*2 is 4, so 4_5. Subtract that from 4_5, which gives 0. Bring down the next digit, which is 1_5. So now we have 01_5.

Now, how many times does 2_5 go into 1_5? Well, 2_5 is 2 in base 10, and 1_5 is 1 in base 10. So 2 goes into 1 zero times. So we write 0 as the next digit in the quotient. Multiply 0 by 2_5, which is 0, subtract that from 1_5, giving 1_5. Bring down the next digit, which is 3_5. So now we have 13_5.

Now, how many times does 2_5 go into 13_5? Let me convert 13_5 to base 10 to make it easier. 1*5 + 3 = 8. And 2_5 is 2 in base 10. So 8 divided by 2 is 4. So 4 times. So we write 4 as the next digit in the quotient. Multiply 4_5 by 2_5. Let me compute that in base 5. 4*2 is 8 in base 10. Now, convert 8 to base 5. 5 goes into 8 once with a remainder of 3. So 8 is 13_5. Therefore, 4_5 * 2_5 = 13_5. Subtract that from 13_5, which gives 0. So there's no remainder.

Therefore, the quotient is 2 0 4_5, which is 204_5. So that matches the previous result. Therefore, the answer is 204_5. So both methods give the same result, so I think that's correct.

But just to be thorough, let me check once more by converting 204_5 back to base 10 and multiplying by 2_5 to see if I get 413_5.

First, 204_5 is 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54 in base 10. Then 54 * 2 = 108. Then converting 108 back to base 5. Let me do that.

Divide 108 by 5: 108 ÷ 5 = 21 with a remainder of 3. Then divide 21 by 5: 21 ÷ 5 = 4 with a remainder of 1. Then divide 4 by 5: 0 with a remainder of 4. So writing the remainders from last to first: 4, 1, 3. Therefore, 108 in base 10 is 413_5. Which matches the original dividend. Perfect! So that confirms that 204_5 multiplied by 2_5 is indeed 413_5. Therefore, the division is correct.

Alternatively, if I wanted to do the division entirely in base 5 without converting to base 10, let me try that again step by step to ensure I didn't make any mistakes in the earlier process.

Starting with 413_5 divided by 2_5.

First, set up the long division:

          ______
2_5 ) 413_5

First, take the first digit of the dividend, which is 4_5. How many times does 2_5 go into 4_5? As before, 2 times. So write 2 above the first digit.

Multiply 2_5 by 2_5. As before, 2*2=4, which is 4_5. Subtract 4_5 from 4_5, get 0. Bring down the next digit, which is 1_5. Now we have 01_5.

How many times does 2_5 go into 1_5? 0 times. So write 0 above the second digit. Multiply 0 by 2_5, which is 0_5. Subtract 0_5 from 1_5, get 1_5. Bring down the next digit, which is 3_5, making it 13_5.

Now, how many times does 2_5 go into 13_5? Let me think in base 5. Let me try multiplying 2_5 by 4_5. 2*4=8 in base 10, which is 13_5 as before. So 4 times. So write 4 above the third digit. Multiply 4_5 by 2_5, which is 13_5. Subtract that from 13_5, get 0. So the division is exact, with quotient 204_5. Yep, same result.

Therefore, regardless of the method, the answer is 204_5. So I think that's solid. But just to make sure, let me try another approach. Maybe using base 5 arithmetic for the division steps.

Alternatively, since 2_5 is 2 in base 10, and 413_5 is 108 in base 10, and 108 divided by 2 is 54, which is 204_5. So that's another way to think about it. All methods converge to the same answer.

Alternatively, maybe I can think of 413_5 divided by 2_5 as multiplying by the reciprocal. But since 2_5 is 2, the reciprocal is 1/2. But in base 5, how would that work? Maybe not necessary here.

Alternatively, maybe I can perform the division step by step in base 5, using base 5 addition and subtraction. Let me try that.

Starting again:

Dividing 413_5 by 2_5.

First, 2_5 into 4_5: 2 times. 2*2=4, subtract, remainder 0. Bring down 1.

Now, 2_5 into 1_5: 0 times. 0*2=0, subtract, remainder 1. Bring down 3, making it 13_5.

Now, 2_5 into 13_5. Let me think how many times. Let me try 2_5 * 1_5 = 2_5. 2_5 * 2_5 = 4_5. 2_5 * 3_5 = 6_5 (which is 11_5). Wait, but 2_5 * 4_5 is 8_10, which is 13_5. So 4 times. So 4_5. So 4*2=8, which is 13_5. Subtract 13_5 from 13_5, remainder 0. So quotient is 2 0 4_5. Yep, same as before.

Therefore, all methods lead to 204_5. Therefore, I can be confident that the answer is 204 base 5.

But just to make sure, let me check if there's any alternative way someone might approach this problem and perhaps make a mistake. For example, if someone tried to do the division without converting, maybe they might miscalculate the multiplication or subtraction steps. But since I've verified through multiple methods, I think it's safe.

Another thing to check: when converting 54 to base 5, I said 5^2 is 25, 5^3 is 125. So 54 divided by 25 is 2, remainder 4. Then 4 divided by 5 is 0, remainder 4. So digits are 2, 0, 4. So 204_5. Correct.

Alternatively, if I didn't remember the conversion steps, I could use division by 5 repeatedly. Let me try that.

Starting with 54:

54 divided by 5 is 10 with remainder 4. Then 10 divided by 5 is 2 with remainder 0. Then 2 divided by 5 is 0 with remainder 2. So writing the remainders from last to first: 2, 0, 4. So 204_5. Yep, same result.

Therefore, all methods confirm that the answer is 204_5. Therefore, I think that's the correct answer.

**Final Answer**
The quotient of $413_5 \div 2_5$ in base 5 is \boxed{204_5}.

To solve the problem of dividing $ 413_5 \div 2_5 $ and expressing the result in base 5, we can follow a systematic approach that involves converting the numbers to base 10, performing the division, and then converting the result back to base 5.

---

### **Step 1: Convert the Dividend from Base 5 to Base 10**

The number $ 413_5 $ is in base 5. To convert it to base 10, we use the place values of base 5:

$$
413_5 = 4 \cdot 5^2 + 1 \cdot 5^1 + 3 \cdot 5^0
$$

$$
= 4 \cdot 25 + 1 \cdot 5 + 3 \cdot 1 = 100 + 5 + 3 = 108_{10}
$$

---

### **Step 2: Convert the Divisor from Base 5 to Base 10**

The number $ 2_5 $ is straightforward:

$$
2_5 = 2 \cdot 5^0 = 2_{10}
$$

---

### **Step 3: Perform the Division in Base 10**

Now divide the base 10 equivalents:

$$
108 \div 2 = 54
$$

---

### **Step 4: Convert the Result Back to Base 5**

We now convert $ 54_{10} $ to base 5. To do this, we divide by 5 and record the remainders:

- $ 54 \div 5 = 10 $ remainder $ 4 $
- $ 10 \div 5 = 2 $ remainder $ 0 $
- $ 2 \div 5 = 0 $ remainder $ 2 $

Reading the remainders from bottom to top gives:

$$
54_{10} = 204_5
$$

---

### **Final Answer**

$$
\boxed{204_5}
$$

[Assistant]
Explanation:
Okay, so I need to divide 413 base 5 by 2 base 5 and give the answer in base 5. Hmm, let me think. First, maybe I should convert both numbers to base 10, do the division there, and then convert the result back to base 5? That might be easier for me since I'm more comfortable with base 10 arithmetic. Let me try that approach.

Starting with the dividend, which is 413 in base 5. To convert that to base 10, I need to remember that each digit represents a power of 5. The rightmost digit is 5^0, then 5^1, then 5^2, and so on. So let's break down 413_5:

The digits are 4, 1, 3 from left to right. So that would be:

4 * 5^2 + 1 * 5^1 + 3 * 5^0.

Calculating each term:

First term: 4 * 5^2. 5 squared is 25, so 4 * 25 = 100.

Second term: 1 * 5^1 = 1 * 5 = 5.

Third term: 3 * 5^0 = 3 * 1 = 3.

Adding them up: 100 + 5 + 3 = 108. So 413 base 5 is 108 in base 10.

Now the divisor is 2_5. Converting that to base 10 is straightforward: 2 * 5^0 = 2 * 1 = 2. So 2 base 5 is 2 in base 10.

So now the problem becomes 108 divided by 2 in base 10. Let me do that division. 108 divided by 2 is 54. So the result in base 10 is 54. Now I need to convert 54 back to base 5.

To convert 54 to base 5, I need to find the coefficients for the powers of 5. Let me recall that 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125. Since 5^3 is 125 which is larger than 54, the highest power needed is 5^2 = 25.

So let's divide 54 by 25 to find the coefficient for 5^2. 54 divided by 25 is 2 with a remainder. 2*25 = 50. So the coefficient for 5^2 is 2. Then subtract 50 from 54, which leaves 4.

Now, moving to the next lower power, which is 5^1 = 5. Take the remainder 4 and divide by 5. That's 0 with a remainder of 4. So the coefficient for 5^1 is 0.

Then the remainder is 4, which is the coefficient for 5^0. So putting it all together, the base 5 number is 2 (for 5^2), 0 (for 5^1), and 4 (for 5^0). Therefore, 54 in base 10 is 204 in base 5.

Wait, let me check that again. Let me verify the conversion of 204 base 5 back to base 10 to make sure. 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Yes, that's correct. So 54 base 10 is indeed 204 base 5. Therefore, the quotient of 413_5 divided by 2_5 is 204_5.

But hold on, maybe I should try doing the division directly in base 5 to confirm? Sometimes converting back and forth can introduce errors, especially if I made a mistake in conversion. Let me try that method as well to verify.

So, let's perform the division 413_5 ÷ 2_5 directly in base 5. Let me recall how division works in different bases. It's similar to base 10, but all the operations are done in base 5.

First, let me write down the dividend and divisor:

Dividend: 413_5

Divisor: 2_5

So, we need to divide 413_5 by 2_5. Let me set it up like long division.

First, I need to see how many times 2_5 goes into the first digit of the dividend. The first digit is 4_5. Let me convert 4_5 to base 10 to understand its value. 4 in base 5 is 4 in base 10. So 4 divided by 2 is 2. So 2_5 goes into 4_5 twice. So I write 2 as the first digit of the quotient.

Multiply 2_5 by 2_5. Let me compute that in base 5. 2*2 is 4, so 4_5. Subtract that from 4_5, which gives 0. Bring down the next digit, which is 1_5. So now we have 01_5.

Now, how many times does 2_5 go into 1_5? Well, 2_5 is 2 in base 10, and 1_5 is 1 in base 10. So 2 goes into 1 zero times. So we write 0 as the next digit in the quotient. Multiply 0 by 2_5, which is 0, subtract that from 1_5, giving 1_5. Bring down the next digit, which is 3_5. So now we have 13_5.

Now, how many times does 2_5 go into 13_5? Let me convert 13_5 to base 10 to make it easier. 1*5 + 3 = 8. And 2_5 is 2 in base 10. So 8 divided by 2 is 4. So 4 times. So we write 4 as the next digit in the quotient. Multiply 4_5 by 2_5. Let me compute that in base 5. 4*2 is 8 in base 10. Now, convert 8 to base 5. 5 goes into 8 once with a remainder of 3. So 8 is 13_5. Therefore, 4_5 * 2_5 = 13_5. Subtract that from 13_5, which gives 0. So there's no remainder.

Therefore, the quotient is 2 0 4_5, which is 204_5. So that matches the previous result. Therefore, the answer is 204_5. So both methods give the same result, so I think that's correct.

But just to be thorough, let me check once more by converting 204_5 back to base 10 and multiplying by 2_5 to see if I get 413_5.

First, 204_5 is 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54 in base 10. Then 54 * 2 = 108. Then converting 108 back to base 5. Let me do that.

Divide 108 by 5: 108 ÷ 5 = 21 with a remainder of 3. Then divide 21 by 5: 21 ÷ 5 = 4 with a remainder of 1. Then divide 4 by 5: 0 with a remainder of 4. So writing the remainders from last to first: 4, 1, 3. Therefore, 108 in base 10 is 413_5. Which matches the original dividend. Perfect! So that confirms that 204_5 multiplied by 2_5 is indeed 413_5. Therefore, the division is correct.

Alternatively, if I wanted to do the division entirely in base 5 without converting to base 10, let me try that again step by step to ensure I didn't make any mistakes in the earlier process.

Starting with 413_5 divided by 2_5.

First, set up the long division:

          ______
2_5 ) 413_5

First, take the first digit of the dividend, which is 4_5. How many times does 2_5 go into 4_5? As before, 2 times. So write 2 above the first digit.

Multiply 2_5 by 2_5. As before, 2*2=4, which is 4_5. Subtract 4_5 from 4_5, get 0. Bring down the next digit, which is 1_5. Now we have 01_5.

How many times does 2_5 go into 1_5? 0 times. So write 0 above the second digit. Multiply 0 by 2_5, which is 0_5. Subtract 0_5 from 1_5, get 1_5. Bring down the next digit, which is 3_5, making it 13_5.

Now, how many times does 2_5 go into 13_5? Let me think in base 5. Let me try multiplying 2_5 by 4_5. 2*4=8 in base 10, which is 13_5 as before. So 4 times. So write 4 above the third digit. Multiply 4_5 by 2_5, which is 13_5. Subtract that from 13_5, get 0. So the division is exact, with quotient 204_5. Yep, same result.

Therefore, regardless of the method, the answer is 204_5. So I think that's solid. But just to make sure, let me try another approach. Maybe using base 5 arithmetic for the division steps.

Alternatively, since 2_5 is 2 in base 10, and 413_5 is 108 in base 10, and 108 divided by 2 is 54, which is 204_5. So that's another way to think about it. All methods converge to the same answer.

Alternatively, maybe I can think of 413_5 divided by 2_5 as multiplying by the reciprocal. But since 2_5 is 2, the reciprocal is 1/2. But in base 5, how would that work? Maybe not necessary here.

Alternatively, maybe I can perform the division step by step in base 5, using base 5 addition and subtraction. Let me try that.

Starting again:

Dividing 413_5 by 2_5.

First, 2_5 into 4_5: 2 times. 2*2=4, subtract, remainder 0. Bring down 1.

Now, 2_5 into 1_5: 0 times. 0*2=0, subtract, remainder 1. Bring down 3, making it 13_5.

Now, 2_5 into 13_5. Let me think how many times. Let me try 2_5 * 1_5 = 2_5. 2_5 * 2_5 = 4_5. 2_5 * 3_5 = 6_5 (which is 11_5). Wait, but 2_5 * 4_5 is 8_10, which is 13_5. So 4 times. So 4_5. So 4*2=8, which is 13_5. Subtract 13_5 from 13_5, remainder 0. So quotient is 2 0 4_5. Yep, same as before.

Therefore, all methods lead to 204_5. Therefore, I can be confident that the answer is 204 base 5.

But just to make sure, let me check if there's any alternative way someone might approach this problem and perhaps make a mistake. For example, if someone tried to do the division without converting, maybe they might miscalculate the multiplication or subtraction steps. But since I've verified through multiple methods, I think it's safe.

Another thing to check: when converting 54 to base 5, I said 5^2 is 25, 5^3 is 125. So 54 divided by 25 is 2, remainder 4. Then 4 divided by 5 is 0, remainder 4. So digits are 2, 0, 4. So 204_5. Correct.

Alternatively, if I didn't remember the conversion steps, I could use division by 5 repeatedly. Let me try that.

Starting with 54:

54 divided by 5 is 10 with remainder 4. Then 10 divided by 5 is 2 with remainder 0. Then 2 divided by 5 is 0 with remainder 2. So writing the remainders from last to first: 2, 0, 4. So 204_5. Yep, same result.

Therefore, all methods confirm that the answer is 204_5. Therefore, I think that's the correct answer.

**Final Answer**
The quotient of $413_5 \div 2_5$ in base 5 is \boxed{204_5}.

To solve the problem of dividing $ 413_5 \div 2_5 $ and expressing the result in base 5, we can follow a systematic approach that involves converting the numbers to base 10, performing the division, and then converting the result back to base 5.

---

### **Step 1: Convert the Dividend from Base 5 to Base 10**

The number $ 413_5 $ is in base 5. To convert it to base 10, we use the place values of base 5:

$$
413_5 = 4 \cdot 5^2 + 1 \cdot 5^1 + 3 \cdot 5^0
$$

$$
= 4 \cdot 25 + 1 \cdot 5 + 3 \cdot 1 = 100 + 5 + 3 = 108_{10}
$$

---

### **Step 2: Convert the Divisor from Base 5 to Base 10**

The number $ 2_5 $ is straightforward:

$$
2_5 = 2 \cdot 5^0 = 2_{10}
$$

---

### **Step 3: Perform the Division in Base 10**

Now divide the base 10 equivalents:

$$
108 \div 2 = 54
$$

---

### **Step 4: Convert the Result Back to Base 5**

We now convert $ 54_{10} $ to base 5. To do this, we divide by 5 and record the remainders:

- $ 54 \div 5 = 10 $ remainder $ 4 $
- $ 10 \div 5 = 2 $ remainder $ 0 $
- $ 2 \div 5 = 0 $ remainder $ 2 $

Reading the remainders from bottom to top gives:

$$
54_{10} = 204_5
$$

---

### **Final Answer**

$$
\boxed{204_5}
$$

Solution:
We may carry out long division in base 5 just as in base 10. We have  \[
\begin{array}{c|ccc}
\multicolumn{2}{r}{2} & 0 & 4 \\
\cline{2-4}
2 & 4 & 1 & 3 \\
\multicolumn{2}{r}{4} & \downarrow & \\ \cline{2-2}
\multicolumn{2}{r}{0} & 1 & \\
\multicolumn{2}{r}{} & 0 & \downarrow \\ \cline{3-3}
\multicolumn{2}{r}{} & 1 & 3 \\
\multicolumn{2}{r}{} & 1 & 3 \\ \cline{3-4}
\multicolumn{2}{r}{} & & 0
\end{array}
\]for a quotient of $\boxed{204_5}$. Note that in the above calculation we have used that $13_5$ divided by $2_5$ is $4_5$, which follows from $4_5\times2_5=8_{10}=13_5$.