Explanation:
Okay, so I need to figure out how many of the same digits are found in the base 7 and base 8 representations of the number 629 in decimal. Let me start by recalling how to convert numbers from base 10 to other bases.

First, let me convert 629 into base 7. To do that, I need to divide 629 by 7 repeatedly and record the remainders. Let me start:

Divide 629 by 7.

7 goes into 629 how many times? Let me calculate 7*89 = 623, which is less than 629. Then 7*90 = 630, which is 1 more than 629. So 629 divided by 7 is 89 with a remainder of 6. So the least significant digit (rightmost digit) is 6.

Now take the quotient 89 and divide by 7. 7*12 = 84, remainder 5. So the next digit is 5.

Then take 12 and divide by 7. 7*1 = 7, remainder 5. Next digit is 5.

Then take 1 and divide by 7. That's 0 with remainder 1. So the next digit is 1.

So reading the remainders from last to first, the base 7 representation is 1 5 5 6. Let me check: 1*7^3 + 5*7^2 + 5*7 + 6 = 343 + 245 + 35 + 6 = 343 + 245 is 588, 588 + 35 is 623, 623 + 6 is 629. Perfect. So base 7 is 1556_7.

Now let me convert 629 to base 8. Again, divide by 8.

629 divided by 8: 8*78 = 624, so 629 - 624 = 5. So first remainder is 5.

Take 78 and divide by 8: 8*9 = 72, remainder 6. Next digit is 6.

Take 9 and divide by 8: 1 with remainder 1. Next digit is 1.

Take 1 and divide by 8: 0 with remainder 1. Next digit is 1.

So reading the remainders from last to first, the base 8 representation is 1 1 6 5, so 1165_8. Let me check: 1*8^3 + 1*8^2 + 6*8 + 5 = 512 + 64 + 48 + 5 = 512 + 64 is 576, 576 + 48 is 624, 624 +5 is 629. Correct. So base 8 is 1165_8.

Now the question is asking how many of the same digits are found in these two representations. So base 7 is 1,5,5,6 and base 8 is 1,1,6,5. Let me list the digits:

Base 7 digits: 1, 5, 5, 6.

Base 8 digits: 1, 1, 6, 5.

Now, we need to find the common digits. But we have to be careful: if a digit appears multiple times in one base, does that count each occurrence? For example, in base 7, there are two 5s, and in base 8, there is one 5. So do we count that as one common digit or two? Wait, the problem says "how many of the same digits are found". Let me check the example given.

The example says that 121_3 and 413_5 have one digit in common. Let me see. 121_3 has digits 1, 2, 1. 413_5 has digits 4,1,3. The common digit is 1. So even though 1 appears twice in the first number, it's only counted once. Therefore, probably, the question is asking for the number of distinct digits that are common between the two representations. So even if a digit is repeated in one base, if it's present in the other, it counts once.

But let me make sure. Let me check the exact wording: "how many of the same digits are found in the base 7 and base 8 representations". So, if there is a digit that is present in both representations, regardless of how many times it appears in each, it counts once. For example, if a digit appears twice in one and once in the other, it's still one digit. Therefore, the answer is the number of distinct digits that are common between the two sets of digits.

Alternatively, if they meant the total number of matching digits, considering multiplicity, then we need to count how many digits are the same in the same position? But no, the example shows that in 121_3 and 413_5, the digits are 1,2,1 and 4,1,3. So they have one common digit, which is 1. But in that case, even though there are two 1s in the first number, but only one 1 in the second. But they are considering the digit 1 as one common digit. Therefore, it's about the set of digits.

Therefore, I think the question wants the number of distinct digits that are present in both representations. So, for example, if base 7 has 1,5,5,6 and base 8 has 1,1,6,5, then the digits in base 7 are {1,5,6} and in base 8 they are {1,5,6}. Therefore, all three digits are common. Therefore, answer is 3? But let me check again.

Wait, base 7 digits: 1, 5, 5, 6. So unique digits are 1,5,6. Base 8 digits: 1,1,6,5. Unique digits are 1,5,6. Therefore, all three digits are common. So the answer is 3?

Wait, but let me check again. Let me make sure that my conversion is correct.

Base 7: 1556. So digits are 1,5,5,6.

Base 8: 1165. So digits are 1,1,6,5. So digits are 1,6,5. So yes, all three digits are common. Therefore, answer is 3? But let me check again with the example given in the problem.

The example says that 121_3 (digits 1,2,1) and 413_5 (digits 4,1,3) have one digit in common. So the digits in first are {1,2}, in second {4,1,3}, so common digit is 1. Therefore, answer is 1. So yeah, the question is about the number of unique digits common to both.

So then, according to that, our case would be three digits. Wait, but let me check again my conversion of 629 into base 7 and base 8.

First base 7: 629 divided by 7: 629 /7= 89.857..., so 89*7=623, 629-623=6. Then 89 /7= 12.714..., 12*7=84, 89-84=5. Then 12 /7=1.714..., 1*7=7, 12-7=5. Then 1 /7=0.142..., remainder 1. So digits from last to first: 1,5,5,6. So 1556_7. Correct.

Base 8: 629 /8= 78.625, so 78*8=624, remainder 5. Then 78 /8=9.75, 9*8=72, remainder 6. Then 9 /8=1.125, remainder 1. Then 1 /8=0, remainder 1. So digits are 1,1,6,5. So 1165_8. Correct. So digits in base 7: 1,5,6. Digits in base 8: 1,5,6. So all three digits are common. Therefore, answer is 3?

But wait, let me check again. Wait, in base 8, the digits are 1, 1, 6, 5. So digits present are 1, 6, 5. In base 7 digits are 1, 5, 6. So same digits. So the digits are 1, 5, 6. Therefore, three digits. So answer is 3?

But wait, the problem says "how many of the same digits are found in the base 7 and base 8 representations". Maybe I need to check if digits are in the same positions? Or maybe not. Let me check the example again.

The example: 121_3 and 413_5. So 121_3 is three digits, 413_5 is three digits. The digits in base 3 are 1, 2, 1 and in base 5 are 4, 1, 3. So the common digit is 1. So even though 1 appears twice in the first, and once in the second, they count as one. So the answer is 1. Therefore, it's about the set of digits.

So in our case, base 7 has digits {1,5,6} and base 8 has digits {1,5,6}. Therefore, they have 3 common digits. Therefore, answer is 3? But wait, maybe I made a mistake here. Wait, is 6 a valid digit in base 7? Yes, because base 7 digits go from 0 to 6. Similarly, base 8 digits go from 0 to 7. So 6 is valid in both. So digits in base 7 are 1, 5, 6. Digits in base 8 are 1, 5, 6. So same digits. Hence, three digits. Therefore, answer is 3.

Wait but let me check if there is any mistake. Wait, is there any digit that is present in both representations? Let me list all digits:

Base 7 digits: 1, 5, 5, 6.

Base 8 digits: 1, 1, 6, 5.

So digits in base 7: 1, 5, 6.

Digits in base 8: 1, 5, 6.

Therefore, all three digits are common. So answer is 3.

But let me check again. Maybe the problem is looking for how many digits are the same in both representations, counting duplicates? For example, if in base 7 there are two 5s and in base 8 there is one 5, then maybe count 1? But according to the example, it's not. Because in the example, base 3 has two 1s and base 5 has one 1, but they count it as one.

Alternatively, maybe the question is not about the set, but about the count of digits that match in both representations, considering their positions. Wait, but the example is different. Let me think.

If the problem had asked about the number of digits that are the same in position, then it would be different. For example, if base 7 is 1556 and base 8 is 1165, then comparing each digit:

First digit: 1 vs 1 (same)

Second digit: 5 vs 1 (different)

Third digit: 5 vs 6 (different)

Fourth digit: 6 vs 5 (different)

So only one digit is same in position. But the example says that 121_3 and 413_5 have one digit in common. If they compared by position, then first digit 1 vs 4 (different), second 2 vs 1 (different), third 1 vs 3 (different). So none same. But in the example, they say there is one digit in common. Hence, it's not about position.

Therefore, the example suggests that it's about the digits themselves, regardless of position, so how many digits are common. So in the example, 1 is common, so answer is 1.

Therefore, in our case, since digits in base 7 are {1,5,6} and in base 8 are {1,5,6}, answer is 3. Therefore, the answer should be 3?

But wait, let me check again. Wait, is 6 present in both? Yes. 5 is present. 1 is present. Therefore, three digits. Hmm. But I need to be 100% sure. Let me check if there are any other digits. For example, is there any other digit? In base 7, digits are 0-6. In base 8, digits are 0-7. The number 629 in base 7 is 1556. So digits are 1,5,5,6. In base 8 is 1165, digits are 1,1,6,5. So no other digits. Therefore, digits are 1,5,6 in both. Hence, 3 digits. Therefore, answer is 3.

But wait, wait, the original question says "how many of the same digits are found in the base 7 and base 8 representations". Maybe the problem is not about the digits present in both representations, but how many digits are same when you look at both representations. For example, if a digit is present in both, count it once. If it is present multiple times in both, still count once. So yes, as per the example. So 3.

But now, maybe I need to check if there's a mistake in my conversion. Because 3 seems straightforward, but maybe I made a mistake? Let me double check.

Let me convert 629 to base 7 again:

Find the largest power of 7 less than 629. 7^0=1, 7^1=7, 7^2=49, 7^3=343, 7^4=2401. So 7^3=343 is less than 629. So let's see:

How many times does 343 go into 629? 629 divided by 343 is 1.83... So coefficient for 7^3 is 1. Then subtract 343 from 629: 629 - 343 = 286.

Next, 7^2 is 49. How many times does 49 go into 286? Let me divide 286 by 49. 49*5=245, 49*6=294 which is too big. So coefficient for 7^2 is 5. Subtract 245 from 286: 286 - 245 = 41.

Next, 7^1 is 7. How many times does 7 go into 41? 7*5=35, 7*6=42. So coefficient is 5. Subtract 35: 41-35=6.

Then 7^0 is 1, coefficient is 6. So digits are 1,5,5,6. So same as before. Correct.

Base 8: Let me do same method.

Powers of 8: 8^0=1, 8^1=8, 8^2=64, 8^3=512, 8^4=4096. So 8^3=512 is less than 629. How many times does 512 go into 629? Once. Coefficient for 8^3 is 1. Subtract 512: 629 - 512 = 117.

Next, 8^2 is 64. How many times does 64 go into 117? 64*1=64, 64*2=128 which is too big. So coefficient is 1. Subtract 64: 117 - 64 = 53.

Next, 8^1 is 8. How many times does 8 go into 53? 8*6=48, 8*7=56. So coefficient 6. Subtract 48: 53 - 48 = 5.

Then 8^0 is 1, coefficient 5. So digits are 1,1,6,5. So 1165_8. Correct.

So digits in base 7: 1,5,5,6. So digits are 1,5,6. In base 8: 1,1,6,5. Digits are 1,5,6. So same digits. Therefore, answer is 3. Therefore, I think the answer is 3. But maybe the problem is trickier?

Wait, let me check if there is any digit that is same but maybe in different positions? But no, the digits themselves are same. For example, in base 7, there is a 1, and in base 8, there is a 1. Similarly for 5 and 6. Therefore, three digits. So answer should be 3.

But maybe the problem counts 0? Wait, but neither base 7 nor base 8 representations of 629 contain 0. Because in base 7: 1556. No zeros. In base 8: 1165. Also no zeros. Therefore, 0 is not present in either representation. Therefore, no.

Wait, another way: Let me write the digits of base 7 and base 8:

Base 7: [1, 5, 5, 6]

Base 8: [1, 1, 6, 5]

Now, the set of digits in base 7 is {1,5,6}; set in base 8 is {1,5,6}. So intersection is {1,5,6}, so size 3.

Alternatively, if the question had meant the digits that are same in the same positions, but as discussed earlier, the example shows that it's not about positions. Therefore, answer is 3.

But maybe I need to check the problem statement again. It says: "how many of the same digits are found in the base 7 and base 8 representations". The wording is a bit ambiguous, but given the example, it's about the digits themselves.

Alternatively, maybe the problem is expecting to count the number of digits that are same in both representations, considering multiplicity? For example, if a digit appears in both, count how many times it appears in both? But that would be different. Let's check.

For example, if in base 7, digit 1 appears once, digit 5 appears twice, digit 6 once. In base 8, digit 1 appears twice, digit 5 once, digit 6 once. So for digit 1: common? How? If we count the minimum of the counts in each base? Like, for 1: 1 in base 7, 2 in base 8, so 1? For 5: 2 in base 7, 1 in base 8, so 1? For 6: 1 in both. So total 1 + 1 +1 = 3? So same answer. Or if we count all occurrences in both? Like for 1: in base 7, 1 occurrence, base 8: 2. So total 1 + 2 = 3? But the example was different. Let me check the example with that approach.

Example: 121_3 (digits 1,2,1) and 413_5 (digits 4,1,3). So digit 1 is present in both. In base 3, two 1s. In base 5, one 1. If we count minimum, it's 1. If we count maximum, it's 2. If we count sum, it's 3. But the example says there is 1 digit in common. Therefore, probably, they consider unique digits. Therefore, the answer is 3.

But just to be thorough, let me check if the problem is from somewhere, maybe it's a known problem. But since I can't do that, I need to rely on my logic.

Alternatively, maybe I misread the question. Wait, the problem says "how many of the same digits are found in the base 7 and base 8 representations". So, for example, if a digit is in both representations, count it once. So, as in the example, even if digit 1 is present multiple times, it's only counted once. So answer is 3.

But just to be 100% sure, let me think of another way. Suppose I list all digits in base 7 and base 8, and count overlaps.

Base 7 digits: 1, 5, 5, 6.

Base 8 digits: 1, 1, 6, 5.

So, digits in base 7: 1,5,5,6.

Digits in base 8: 1,1,6,5.

Now, how many digits are common? Let me compare each digit in base 7 with each digit in base 8. But since digits can be repeated, but we need to count unique digits. So:

Digits in base 7: 1,5,6.

Digits in base 8: 1,5,6.

So, intersection is 3. Hence, answer is 3. So I think that is correct.

Alternatively, if the problem had said "how many digits are the same when you write them out", considering repetitions, but given the example, that's not the case. Therefore, answer is 3.

But to be extra cautious, let me check with another example. Let's say if a number in base 7 was 111_7 and base 8 was 111_8. Then digits are all 1s. So same digits, so answer would be 1? Or 3? According to the example, it should be 1. Because they are same digits. So even though there are three 1s, they are the same digit, so count once. Hence, answer is 1. Therefore, the answer should be number of unique digits common to both. Hence, 3 in our case.

Therefore, I think the answer is 3. Therefore, \boxed{3}.

Wait, but I just thought, maybe the original number 629 has some digits that are same in both representations? Wait, but digits are 1,5,6 in both. So three digits. Therefore, answer is 3.

But just to check, maybe there is a mistake in my base conversions. Let me check once again.

For base 7: 1*7^3 + 5*7^2 + 5*7 + 6 = 1*343 + 5*49 + 5*7 +6 = 343 + 245 + 35 + 6 = 343 + 245 = 588; 588 + 35 = 623; 623 + 6 = 629. Correct.

For base 8: 1*8^3 + 1*8^2 + 6*8 +5 = 512 + 64 + 48 + 5 = 512 + 64 is 576, 576 + 48 is 624, 624 + 5 is 629. Correct. So no mistake here.

Therefore, I think my conclusion is correct. The answer is 3. Therefore, the number of same digits is 3.

**Final Answer**
\boxed{3}

To determine how many of the same digits are found in the base 7 and base 8 representations of the number $ 629_{10} $, we begin by converting the number from base 10 to both base 7 and base 8.

---

### **Step 1: Convert 629 from base 10 to base 7**

We do this by successive division:

- $ 629 \div 7 = 89 $ remainder **6**
- $ 89 \div 7 = 12 $ remainder **5**
- $ 12 \div 7 = 1 $ remainder **5**
- $ 1 \div 7 = 0 $ remainder **1**

Reading the remainders from last to first, we get:

$$
629_{10} = 1556_7
$$

Digits in base 7: **1, 5, 5, 6**

---

### **Step 2: Convert 629 from base 10 to base 8**

Again, we perform successive division:

- $ 629 \div 8 = 78 $ remainder **5**
- $ 78 \div 8 = 9 $ remainder **6**
- $ 9 \div 8 = 1 $ remainder **1**
- $ 1 \div 8 = 0 $ remainder **1**

Reading the remainders from last to first, we get:

$$
629_{10} = 1165_8
$$

Digits in base 8: **1, 1, 6, 5**

---

### **Step 3: Compare the Digits**

Now we find the **distinct** digits present in both base 7 and base 8 representations:

- **Base 7 digits:** {1, 5, 6}
- **Base 8 digits:** {1, 5, 6}

We observe that all three digits (1, 5, and 6) are present in both representations. Note that although some digits might appear multiple times in a representation, we count each **distinct digit** only once.

---

### **Conclusion**

The number of **distinct digits** that are common to both the base 7 and base 8 representations of 629 is:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many of the same digits are found in the base 7 and base 8 representations of the number 629 in decimal. Let me start by recalling how to convert numbers from base 10 to other bases.

First, let me convert 629 into base 7. To do that, I need to divide 629 by 7 repeatedly and record the remainders. Let me start:

Divide 629 by 7.

7 goes into 629 how many times? Let me calculate 7*89 = 623, which is less than 629. Then 7*90 = 630, which is 1 more than 629. So 629 divided by 7 is 89 with a remainder of 6. So the least significant digit (rightmost digit) is 6.

Now take the quotient 89 and divide by 7. 7*12 = 84, remainder 5. So the next digit is 5.

Then take 12 and divide by 7. 7*1 = 7, remainder 5. Next digit is 5.

Then take 1 and divide by 7. That's 0 with remainder 1. So the next digit is 1.

So reading the remainders from last to first, the base 7 representation is 1 5 5 6. Let me check: 1*7^3 + 5*7^2 + 5*7 + 6 = 343 + 245 + 35 + 6 = 343 + 245 is 588, 588 + 35 is 623, 623 + 6 is 629. Perfect. So base 7 is 1556_7.

Now let me convert 629 to base 8. Again, divide by 8.

629 divided by 8: 8*78 = 624, so 629 - 624 = 5. So first remainder is 5.

Take 78 and divide by 8: 8*9 = 72, remainder 6. Next digit is 6.

Take 9 and divide by 8: 1 with remainder 1. Next digit is 1.

Take 1 and divide by 8: 0 with remainder 1. Next digit is 1.

So reading the remainders from last to first, the base 8 representation is 1 1 6 5, so 1165_8. Let me check: 1*8^3 + 1*8^2 + 6*8 + 5 = 512 + 64 + 48 + 5 = 512 + 64 is 576, 576 + 48 is 624, 624 +5 is 629. Correct. So base 8 is 1165_8.

Now the question is asking how many of the same digits are found in these two representations. So base 7 is 1,5,5,6 and base 8 is 1,1,6,5. Let me list the digits:

Base 7 digits: 1, 5, 5, 6.

Base 8 digits: 1, 1, 6, 5.

Now, we need to find the common digits. But we have to be careful: if a digit appears multiple times in one base, does that count each occurrence? For example, in base 7, there are two 5s, and in base 8, there is one 5. So do we count that as one common digit or two? Wait, the problem says "how many of the same digits are found". Let me check the example given.

The example says that 121_3 and 413_5 have one digit in common. Let me see. 121_3 has digits 1, 2, 1. 413_5 has digits 4,1,3. The common digit is 1. So even though 1 appears twice in the first number, it's only counted once. Therefore, probably, the question is asking for the number of distinct digits that are common between the two representations. So even if a digit is repeated in one base, if it's present in the other, it counts once.

But let me make sure. Let me check the exact wording: "how many of the same digits are found in the base 7 and base 8 representations". So, if there is a digit that is present in both representations, regardless of how many times it appears in each, it counts once. For example, if a digit appears twice in one and once in the other, it's still one digit. Therefore, the answer is the number of distinct digits that are common between the two sets of digits.

Alternatively, if they meant the total number of matching digits, considering multiplicity, then we need to count how many digits are the same in the same position? But no, the example shows that in 121_3 and 413_5, the digits are 1,2,1 and 4,1,3. So they have one common digit, which is 1. But in that case, even though there are two 1s in the first number, but only one 1 in the second. But they are considering the digit 1 as one common digit. Therefore, it's about the set of digits.

Therefore, I think the question wants the number of distinct digits that are present in both representations. So, for example, if base 7 has 1,5,5,6 and base 8 has 1,1,6,5, then the digits in base 7 are {1,5,6} and in base 8 they are {1,5,6}. Therefore, all three digits are common. Therefore, answer is 3? But let me check again.

Wait, base 7 digits: 1, 5, 5, 6. So unique digits are 1,5,6. Base 8 digits: 1,1,6,5. Unique digits are 1,5,6. Therefore, all three digits are common. So the answer is 3?

Wait, but let me check again. Let me make sure that my conversion is correct.

Base 7: 1556. So digits are 1,5,5,6.

Base 8: 1165. So digits are 1,1,6,5. So digits are 1,6,5. So yes, all three digits are common. Therefore, answer is 3? But let me check again with the example given in the problem.

The example says that 121_3 (digits 1,2,1) and 413_5 (digits 4,1,3) have one digit in common. So the digits in first are {1,2}, in second {4,1,3}, so common digit is 1. Therefore, answer is 1. So yeah, the question is about the number of unique digits common to both.

So then, according to that, our case would be three digits. Wait, but let me check again my conversion of 629 into base 7 and base 8.

First base 7: 629 divided by 7: 629 /7= 89.857..., so 89*7=623, 629-623=6. Then 89 /7= 12.714..., 12*7=84, 89-84=5. Then 12 /7=1.714..., 1*7=7, 12-7=5. Then 1 /7=0.142..., remainder 1. So digits from last to first: 1,5,5,6. So 1556_7. Correct.

Base 8: 629 /8= 78.625, so 78*8=624, remainder 5. Then 78 /8=9.75, 9*8=72, remainder 6. Then 9 /8=1.125, remainder 1. Then 1 /8=0, remainder 1. So digits are 1,1,6,5. So 1165_8. Correct. So digits in base 7: 1,5,6. Digits in base 8: 1,5,6. So all three digits are common. Therefore, answer is 3?

But wait, let me check again. Wait, in base 8, the digits are 1, 1, 6, 5. So digits present are 1, 6, 5. In base 7 digits are 1, 5, 6. So same digits. So the digits are 1, 5, 6. Therefore, three digits. So answer is 3?

But wait, the problem says "how many of the same digits are found in the base 7 and base 8 representations". Maybe I need to check if digits are in the same positions? Or maybe not. Let me check the example again.

The example: 121_3 and 413_5. So 121_3 is three digits, 413_5 is three digits. The digits in base 3 are 1, 2, 1 and in base 5 are 4, 1, 3. So the common digit is 1. So even though 1 appears twice in the first, and once in the second, they count as one. So the answer is 1. Therefore, it's about the set of digits.

So in our case, base 7 has digits {1,5,6} and base 8 has digits {1,5,6}. Therefore, they have 3 common digits. Therefore, answer is 3? But wait, maybe I made a mistake here. Wait, is 6 a valid digit in base 7? Yes, because base 7 digits go from 0 to 6. Similarly, base 8 digits go from 0 to 7. So 6 is valid in both. So digits in base 7 are 1, 5, 6. Digits in base 8 are 1, 5, 6. So same digits. Hence, three digits. Therefore, answer is 3.

Wait but let me check if there is any mistake. Wait, is there any digit that is present in both representations? Let me list all digits:

Base 7 digits: 1, 5, 5, 6.

Base 8 digits: 1, 1, 6, 5.

So digits in base 7: 1, 5, 6.

Digits in base 8: 1, 5, 6.

Therefore, all three digits are common. So answer is 3.

But let me check again. Maybe the problem is looking for how many digits are the same in both representations, counting duplicates? For example, if in base 7 there are two 5s and in base 8 there is one 5, then maybe count 1? But according to the example, it's not. Because in the example, base 3 has two 1s and base 5 has one 1, but they count it as one.

Alternatively, maybe the question is not about the set, but about the count of digits that match in both representations, considering their positions. Wait, but the example is different. Let me think.

If the problem had asked about the number of digits that are the same in position, then it would be different. For example, if base 7 is 1556 and base 8 is 1165, then comparing each digit:

First digit: 1 vs 1 (same)

Second digit: 5 vs 1 (different)

Third digit: 5 vs 6 (different)

Fourth digit: 6 vs 5 (different)

So only one digit is same in position. But the example says that 121_3 and 413_5 have one digit in common. If they compared by position, then first digit 1 vs 4 (different), second 2 vs 1 (different), third 1 vs 3 (different). So none same. But in the example, they say there is one digit in common. Hence, it's not about position.

Therefore, the example suggests that it's about the digits themselves, regardless of position, so how many digits are common. So in the example, 1 is common, so answer is 1.

Therefore, in our case, since digits in base 7 are {1,5,6} and in base 8 are {1,5,6}, answer is 3. Therefore, the answer should be 3?

But wait, let me check again. Wait, is 6 present in both? Yes. 5 is present. 1 is present. Therefore, three digits. Hmm. But I need to be 100% sure. Let me check if there are any other digits. For example, is there any other digit? In base 7, digits are 0-6. In base 8, digits are 0-7. The number 629 in base 7 is 1556. So digits are 1,5,5,6. In base 8 is 1165, digits are 1,1,6,5. So no other digits. Therefore, digits are 1,5,6 in both. Hence, 3 digits. Therefore, answer is 3.

But wait, wait, the original question says "how many of the same digits are found in the base 7 and base 8 representations". Maybe the problem is not about the digits present in both representations, but how many digits are same when you look at both representations. For example, if a digit is present in both, count it once. If it is present multiple times in both, still count once. So yes, as per the example. So 3.

But now, maybe I need to check if there's a mistake in my conversion. Because 3 seems straightforward, but maybe I made a mistake? Let me double check.

Let me convert 629 to base 7 again:

Find the largest power of 7 less than 629. 7^0=1, 7^1=7, 7^2=49, 7^3=343, 7^4=2401. So 7^3=343 is less than 629. So let's see:

How many times does 343 go into 629? 629 divided by 343 is 1.83... So coefficient for 7^3 is 1. Then subtract 343 from 629: 629 - 343 = 286.

Next, 7^2 is 49. How many times does 49 go into 286? Let me divide 286 by 49. 49*5=245, 49*6=294 which is too big. So coefficient for 7^2 is 5. Subtract 245 from 286: 286 - 245 = 41.

Next, 7^1 is 7. How many times does 7 go into 41? 7*5=35, 7*6=42. So coefficient is 5. Subtract 35: 41-35=6.

Then 7^0 is 1, coefficient is 6. So digits are 1,5,5,6. So same as before. Correct.

Base 8: Let me do same method.

Powers of 8: 8^0=1, 8^1=8, 8^2=64, 8^3=512, 8^4=4096. So 8^3=512 is less than 629. How many times does 512 go into 629? Once. Coefficient for 8^3 is 1. Subtract 512: 629 - 512 = 117.

Next, 8^2 is 64. How many times does 64 go into 117? 64*1=64, 64*2=128 which is too big. So coefficient is 1. Subtract 64: 117 - 64 = 53.

Next, 8^1 is 8. How many times does 8 go into 53? 8*6=48, 8*7=56. So coefficient 6. Subtract 48: 53 - 48 = 5.

Then 8^0 is 1, coefficient 5. So digits are 1,1,6,5. So 1165_8. Correct.

So digits in base 7: 1,5,5,6. So digits are 1,5,6. In base 8: 1,1,6,5. Digits are 1,5,6. So same digits. Therefore, answer is 3. Therefore, I think the answer is 3. But maybe the problem is trickier?

Wait, let me check if there is any digit that is same but maybe in different positions? But no, the digits themselves are same. For example, in base 7, there is a 1, and in base 8, there is a 1. Similarly for 5 and 6. Therefore, three digits. So answer should be 3.

But maybe the problem counts 0? Wait, but neither base 7 nor base 8 representations of 629 contain 0. Because in base 7: 1556. No zeros. In base 8: 1165. Also no zeros. Therefore, 0 is not present in either representation. Therefore, no.

Wait, another way: Let me write the digits of base 7 and base 8:

Base 7: [1, 5, 5, 6]

Base 8: [1, 1, 6, 5]

Now, the set of digits in base 7 is {1,5,6}; set in base 8 is {1,5,6}. So intersection is {1,5,6}, so size 3.

Alternatively, if the question had meant the digits that are same in the same positions, but as discussed earlier, the example shows that it's not about positions. Therefore, answer is 3.

But maybe I need to check the problem statement again. It says: "how many of the same digits are found in the base 7 and base 8 representations". The wording is a bit ambiguous, but given the example, it's about the digits themselves.

Alternatively, maybe the problem is expecting to count the number of digits that are same in both representations, considering multiplicity? For example, if a digit appears in both, count how many times it appears in both? But that would be different. Let's check.

For example, if in base 7, digit 1 appears once, digit 5 appears twice, digit 6 once. In base 8, digit 1 appears twice, digit 5 once, digit 6 once. So for digit 1: common? How? If we count the minimum of the counts in each base? Like, for 1: 1 in base 7, 2 in base 8, so 1? For 5: 2 in base 7, 1 in base 8, so 1? For 6: 1 in both. So total 1 + 1 +1 = 3? So same answer. Or if we count all occurrences in both? Like for 1: in base 7, 1 occurrence, base 8: 2. So total 1 + 2 = 3? But the example was different. Let me check the example with that approach.

Example: 121_3 (digits 1,2,1) and 413_5 (digits 4,1,3). So digit 1 is present in both. In base 3, two 1s. In base 5, one 1. If we count minimum, it's 1. If we count maximum, it's 2. If we count sum, it's 3. But the example says there is 1 digit in common. Therefore, probably, they consider unique digits. Therefore, the answer is 3.

But just to be thorough, let me check if the problem is from somewhere, maybe it's a known problem. But since I can't do that, I need to rely on my logic.

Alternatively, maybe I misread the question. Wait, the problem says "how many of the same digits are found in the base 7 and base 8 representations". So, for example, if a digit is in both representations, count it once. So, as in the example, even if digit 1 is present multiple times, it's only counted once. So answer is 3.

But just to be 100% sure, let me think of another way. Suppose I list all digits in base 7 and base 8, and count overlaps.

Base 7 digits: 1, 5, 5, 6.

Base 8 digits: 1, 1, 6, 5.

So, digits in base 7: 1,5,5,6.

Digits in base 8: 1,1,6,5.

Now, how many digits are common? Let me compare each digit in base 7 with each digit in base 8. But since digits can be repeated, but we need to count unique digits. So:

Digits in base 7: 1,5,6.

Digits in base 8: 1,5,6.

So, intersection is 3. Hence, answer is 3. So I think that is correct.

Alternatively, if the problem had said "how many digits are the same when you write them out", considering repetitions, but given the example, that's not the case. Therefore, answer is 3.

But to be extra cautious, let me check with another example. Let's say if a number in base 7 was 111_7 and base 8 was 111_8. Then digits are all 1s. So same digits, so answer would be 1? Or 3? According to the example, it should be 1. Because they are same digits. So even though there are three 1s, they are the same digit, so count once. Hence, answer is 1. Therefore, the answer should be number of unique digits common to both. Hence, 3 in our case.

Therefore, I think the answer is 3. Therefore, \boxed{3}.

Wait, but I just thought, maybe the original number 629 has some digits that are same in both representations? Wait, but digits are 1,5,6 in both. So three digits. Therefore, answer is 3.

But just to check, maybe there is a mistake in my base conversions. Let me check once again.

For base 7: 1*7^3 + 5*7^2 + 5*7 + 6 = 1*343 + 5*49 + 5*7 +6 = 343 + 245 + 35 + 6 = 343 + 245 = 588; 588 + 35 = 623; 623 + 6 = 629. Correct.

For base 8: 1*8^3 + 1*8^2 + 6*8 +5 = 512 + 64 + 48 + 5 = 512 + 64 is 576, 576 + 48 is 624, 624 + 5 is 629. Correct. So no mistake here.

Therefore, I think my conclusion is correct. The answer is 3. Therefore, the number of same digits is 3.

**Final Answer**
\boxed{3}

To determine how many of the same digits are found in the base 7 and base 8 representations of the number $ 629_{10} $, we begin by converting the number from base 10 to both base 7 and base 8.

---

### **Step 1: Convert 629 from base 10 to base 7**

We do this by successive division:

- $ 629 \div 7 = 89 $ remainder **6**
- $ 89 \div 7 = 12 $ remainder **5**
- $ 12 \div 7 = 1 $ remainder **5**
- $ 1 \div 7 = 0 $ remainder **1**

Reading the remainders from last to first, we get:

$$
629_{10} = 1556_7
$$

Digits in base 7: **1, 5, 5, 6**

---

### **Step 2: Convert 629 from base 10 to base 8**

Again, we perform successive division:

- $ 629 \div 8 = 78 $ remainder **5**
- $ 78 \div 8 = 9 $ remainder **6**
- $ 9 \div 8 = 1 $ remainder **1**
- $ 1 \div 8 = 0 $ remainder **1**

Reading the remainders from last to first, we get:

$$
629_{10} = 1165_8
$$

Digits in base 8: **1, 1, 6, 5**

---

### **Step 3: Compare the Digits**

Now we find the **distinct** digits present in both base 7 and base 8 representations:

- **Base 7 digits:** {1, 5, 6}
- **Base 8 digits:** {1, 5, 6}

We observe that all three digits (1, 5, and 6) are present in both representations. Note that although some digits might appear multiple times in a representation, we count each **distinct digit** only once.

---

### **Conclusion**

The number of **distinct digits** that are common to both the base 7 and base 8 representations of 629 is:

$$
\boxed{3}
$$

Solution:
First, let us convert $629_{10}$ to each of the two bases. To convert to base 7, we realize $7^{4}>629_{10}>7^{3}$. So, we can tell that $629_{10}$ in base seven will have four digits. $7^{3}=343$, which can go into 629 only one time at most, leaving $629-1\cdot343 = 286$ for the next three digits. $7^{2}=49$ goes into 286 five times at most, leaving us with $286-5\cdot49 = 41$. Then, $7^{1}=7$ goes into 41 five times at most, leaving $41-5\cdot7 = 6$ for the ones digit. All together, the base seven equivalent of $629_{10}$ is $1556_{7}$.

To convert to base 8, we realize similarly that $8^{4}>629_{10}>8^{3}$. So, we can tell that $629_{10}$ in base eight will have four digits. $8^{3}=512$, which can go into 629 only one time at most, leaving $629-1\cdot512 = 117$ for the next three digits. $8^{2}=64$ goes into 117 one time at most, leaving us with $117-1\cdot64 = 53$. Then, $8^{1}=8$ goes into 53 six times at most, leaving $53-6\cdot8 = 5$ for the ones digit. All together, the base eight equivalent of $629_{10}$ is $1165_{8}$.

Finally, comparing $1556_{7}$ and $1165_{8}$, we find that digits 1, 5, and 6 are present in both numbers.  Thus, there are $\boxed{3}$ digits in common.


Explanation:
Okay, so I need to figure out how many ways 3 people can be assigned to 5 different offices if no one shares an office. Each person gets exactly one office. Hmm, let me think about this step by step.

First, let me make sure I understand the problem correctly. We have 5 offices, and 3 people need to be assigned to them. Each person must have their own office, so no two people can be in the same office. That means we need to assign each person to a different office. Right?

So, maybe this is a permutation problem? Or combinations? Wait, since the offices are different, the order might matter? Let me think. Let's say we have offices labeled 1, 2, 3, 4, 5. Assigning person A to office 1 and person B to office 2 is different from assigning person B to office 1 and person A to office 2. So, the order in which we assign people to offices does matter. Therefore, maybe permutations are involved here?

Alternatively, maybe it's combinations? Wait, combinations don't consider order, but here each person is distinct and the offices are distinct. So, perhaps it's permutations. But let me think again.

Alternatively, maybe we can think of this as assigning each person to an office, considering that each office can only be assigned once. So, for the first person, there are 5 choices. Then, for the second person, since one office is already taken, there are 4 choices left. For the third person, there are 3 choices left. So, total number of ways would be 5 * 4 * 3. Let me check.

Yes, that seems right. For the first person, there are 5 offices. Once the first person is assigned to an office, the second person has 4 remaining offices. Then the third person has 3. So, multiplying these together gives 5 * 4 * 3. Which is 60. But wait, let me think again if there's another way.

Alternatively, we can think of selecting 3 offices out of 5 and then arranging the 3 people in those offices. So, first, choose 3 offices from 5. The number of ways to choose 3 offices from 5 is combination C(5,3). Which is calculated as 5! / (3! * (5-3)!) = 10. Then, for each of these 10 combinations of offices, we can assign the 3 people to the offices. Since each person is different, the number of ways to assign them is 3! = 6. So total number of ways is 10 * 6 = 60. Which matches the previous result.

So, both methods give 60. Therefore, the answer is 60?

Alternatively, let me think if there's a different approach. For example, using permutations. The number of permutations of 5 offices taken 3 at a time is P(5,3) which is calculated as 5! / (5-3)! = 5! / 2! = 120 / 2 = 60. Yep, same answer. So that's another way to see it. So, permutations.

So, the answer is 60. Wait, but let me make sure there isn't a trick here. The problem says "no one shares an office", so each person has their own office, which we accounted for by considering permutations where each office is unique. So, yes, each person is assigned to a different office. So, that's correct.

Alternatively, if the problem had said that the offices are not different, like if they were just rooms without labels, then maybe combinations would come into play. But since the offices are different, the assignments are different based on which office each person is in. So, for example, assigning person A to office 1 and person B to office 2 is different from assigning person A to office 2 and person B to office 1. Therefore, permutations.

Alternatively, if the problem had said that the offices are identical, then maybe it's combinations. But since the offices are different, permutations. So, yeah, 5 * 4 * 3 = 60. So, the answer is 60.

Wait, but let me just think of a smaller case to verify. Suppose there are 2 people and 2 offices. Then the number of ways should be 2 * 1 = 2. Alternatively, C(2,2) * 2! = 1 * 2 = 2. Which is correct. If there are 3 people and 3 offices, then it's 3! = 6. Which is correct. So, seems like the formula P(n, k) = n! / (n - k)! gives the number of ways to assign k people to n offices with no sharing. Therefore, in our case, n=5, k=3. So P(5,3) = 5! / 2! = 120 / 2 = 60. So that's correct.

Alternatively, if I think in terms of multiplication principle. The first person can choose any of the 5 offices. Once that is chosen, the second person can choose any of the remaining 4 offices. The third person can choose any of the remaining 3 offices. Therefore, the total number of ways is 5 * 4 * 3 = 60. So same answer.

Therefore, I think 60 is the correct answer here. Let me check once more if there's a different interpretation. For example, maybe "assigned to 5 different offices" is confusing. But if no one shares an office, then each person is in a different office. So, since there are 5 offices, and only 3 people, there are 2 offices that remain unassigned. But the problem says "assigned to 5 different offices", but since only 3 people are assigned, perhaps that wording is just saying that each person is assigned to an office, and the offices are different. Maybe that wording is just emphasizing that the offices are different. So, yeah, probably the answer is 60.

Alternatively, if the problem had said that each office can have at most one person, which would be equivalent. But since the problem says "no one shares an office", which is the same as each office has at most one person, since more than one person in an office would be sharing. So, yeah, that's the same condition as assigning each person to a different office.

Therefore, conclusion: 5 * 4 * 3 = 60. So, the answer is 60. Therefore, \boxed{60} is the answer.

**Final Answer**
The number of ways is \boxed{60}.

To determine the number of ways 3 people can be assigned to 5 different offices, with the condition that **no one shares an office**, let’s break down the problem logically.

---

### Understanding the Problem

We are given:
- **3 people**, and
- **5 office spaces**, each of which is **distinct** (i.e., labeled or otherwise unique).

Each person is to be **assigned exactly one office**, and **no two people may share the same office**. Therefore, we are choosing **3 different offices** out of the 5 to assign to the 3 people, and **assigning the people to those offices** in different ways.

---

### Step 1: Select 3 Offices out of 5

First, we need to find out how many ways we can choose 3 distinct offices from 5. This is a **combinations** problem, because we are simply selecting which offices will be used, without regard to the order.

$$
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2} = 10
$$

So, there are **10** ways to choose 3 distinct offices from the 5 available.

---

### Step 2: Assign the 3 People to the Chosen 3 Offices

Once we have 3 offices, we need to assign 3 **distinct people** to them. Since the offices are **distinct**, switching a person into office A and another into office B counts as a **different arrangement**.

Therefore, the number of ways to assign 3 people to 3 offices is the number of **permutations** of 3 items, which is:

$$
3! = 3 \times 2 \times 1 = 6
$$

---

### Step 3: Multiply the Two Results

We now multiply the number of ways to choose the offices by the number of ways to assign the people:

$$
\text{Total Ways} = \binom{5}{3} \times 3! = 10 \times 6 = 60
$$

---

### Alternative Approach: Using Permutations Directly

Another way to approach the problem is by recognizing that we are **arranging 3 people into 5 distinct offices**, such that each person gets exactly one unique office. This is exactly the definition of **permutations of 5 things taken 3 at a time**, denoted as:

$$
P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} = \frac{120}{2} = 60
$$

So, this confirms the same outcome.

---

### Final Answer

$$
\boxed{60}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways 3 people can be assigned to 5 different offices if no one shares an office. Each person gets exactly one office. Hmm, let me think about this step by step.

First, let me make sure I understand the problem correctly. We have 5 offices, and 3 people need to be assigned to them. Each person must have their own office, so no two people can be in the same office. That means we need to assign each person to a different office. Right?

So, maybe this is a permutation problem? Or combinations? Wait, since the offices are different, the order might matter? Let me think. Let's say we have offices labeled 1, 2, 3, 4, 5. Assigning person A to office 1 and person B to office 2 is different from assigning person B to office 1 and person A to office 2. So, the order in which we assign people to offices does matter. Therefore, maybe permutations are involved here?

Alternatively, maybe it's combinations? Wait, combinations don't consider order, but here each person is distinct and the offices are distinct. So, perhaps it's permutations. But let me think again.

Alternatively, maybe we can think of this as assigning each person to an office, considering that each office can only be assigned once. So, for the first person, there are 5 choices. Then, for the second person, since one office is already taken, there are 4 choices left. For the third person, there are 3 choices left. So, total number of ways would be 5 * 4 * 3. Let me check.

Yes, that seems right. For the first person, there are 5 offices. Once the first person is assigned to an office, the second person has 4 remaining offices. Then the third person has 3. So, multiplying these together gives 5 * 4 * 3. Which is 60. But wait, let me think again if there's another way.

Alternatively, we can think of selecting 3 offices out of 5 and then arranging the 3 people in those offices. So, first, choose 3 offices from 5. The number of ways to choose 3 offices from 5 is combination C(5,3). Which is calculated as 5! / (3! * (5-3)!) = 10. Then, for each of these 10 combinations of offices, we can assign the 3 people to the offices. Since each person is different, the number of ways to assign them is 3! = 6. So total number of ways is 10 * 6 = 60. Which matches the previous result.

So, both methods give 60. Therefore, the answer is 60?

Alternatively, let me think if there's a different approach. For example, using permutations. The number of permutations of 5 offices taken 3 at a time is P(5,3) which is calculated as 5! / (5-3)! = 5! / 2! = 120 / 2 = 60. Yep, same answer. So that's another way to see it. So, permutations.

So, the answer is 60. Wait, but let me make sure there isn't a trick here. The problem says "no one shares an office", so each person has their own office, which we accounted for by considering permutations where each office is unique. So, yes, each person is assigned to a different office. So, that's correct.

Alternatively, if the problem had said that the offices are not different, like if they were just rooms without labels, then maybe combinations would come into play. But since the offices are different, the assignments are different based on which office each person is in. So, for example, assigning person A to office 1 and person B to office 2 is different from assigning person A to office 2 and person B to office 1. Therefore, permutations.

Alternatively, if the problem had said that the offices are identical, then maybe it's combinations. But since the offices are different, permutations. So, yeah, 5 * 4 * 3 = 60. So, the answer is 60.

Wait, but let me just think of a smaller case to verify. Suppose there are 2 people and 2 offices. Then the number of ways should be 2 * 1 = 2. Alternatively, C(2,2) * 2! = 1 * 2 = 2. Which is correct. If there are 3 people and 3 offices, then it's 3! = 6. Which is correct. So, seems like the formula P(n, k) = n! / (n - k)! gives the number of ways to assign k people to n offices with no sharing. Therefore, in our case, n=5, k=3. So P(5,3) = 5! / 2! = 120 / 2 = 60. So that's correct.

Alternatively, if I think in terms of multiplication principle. The first person can choose any of the 5 offices. Once that is chosen, the second person can choose any of the remaining 4 offices. The third person can choose any of the remaining 3 offices. Therefore, the total number of ways is 5 * 4 * 3 = 60. So same answer.

Therefore, I think 60 is the correct answer here. Let me check once more if there's a different interpretation. For example, maybe "assigned to 5 different offices" is confusing. But if no one shares an office, then each person is in a different office. So, since there are 5 offices, and only 3 people, there are 2 offices that remain unassigned. But the problem says "assigned to 5 different offices", but since only 3 people are assigned, perhaps that wording is just saying that each person is assigned to an office, and the offices are different. Maybe that wording is just emphasizing that the offices are different. So, yeah, probably the answer is 60.

Alternatively, if the problem had said that each office can have at most one person, which would be equivalent. But since the problem says "no one shares an office", which is the same as each office has at most one person, since more than one person in an office would be sharing. So, yeah, that's the same condition as assigning each person to a different office.

Therefore, conclusion: 5 * 4 * 3 = 60. So, the answer is 60. Therefore, \boxed{60} is the answer.

**Final Answer**
The number of ways is \boxed{60}.

To determine the number of ways 3 people can be assigned to 5 different offices, with the condition that **no one shares an office**, let’s break down the problem logically.

---

### Understanding the Problem

We are given:
- **3 people**, and
- **5 office spaces**, each of which is **distinct** (i.e., labeled or otherwise unique).

Each person is to be **assigned exactly one office**, and **no two people may share the same office**. Therefore, we are choosing **3 different offices** out of the 5 to assign to the 3 people, and **assigning the people to those offices** in different ways.

---

### Step 1: Select 3 Offices out of 5

First, we need to find out how many ways we can choose 3 distinct offices from 5. This is a **combinations** problem, because we are simply selecting which offices will be used, without regard to the order.

$$
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2} = 10
$$

So, there are **10** ways to choose 3 distinct offices from the 5 available.

---

### Step 2: Assign the 3 People to the Chosen 3 Offices

Once we have 3 offices, we need to assign 3 **distinct people** to them. Since the offices are **distinct**, switching a person into office A and another into office B counts as a **different arrangement**.

Therefore, the number of ways to assign 3 people to 3 offices is the number of **permutations** of 3 items, which is:

$$
3! = 3 \times 2 \times 1 = 6
$$

---

### Step 3: Multiply the Two Results

We now multiply the number of ways to choose the offices by the number of ways to assign the people:

$$
\text{Total Ways} = \binom{5}{3} \times 3! = 10 \times 6 = 60
$$

---

### Alternative Approach: Using Permutations Directly

Another way to approach the problem is by recognizing that we are **arranging 3 people into 5 distinct offices**, such that each person gets exactly one unique office. This is exactly the definition of **permutations of 5 things taken 3 at a time**, denoted as:

$$
P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} = \frac{120}{2} = 60
$$

So, this confirms the same outcome.

---

### Final Answer

$$
\boxed{60}
$$

Solution: If we number the three people 1, 2, and 3, there are $5$ offices that person 1 can be assigned to, $4$ offices that person 2 can be assigned to, and $3$ offices that person 3 can be assigned to. This gives us $5 \times 4 \times 3 = \boxed{60}$ ways to assign the three people to offices.


Explanation:
Okay, so I need to find the value of the inverse function of f at 1/5, which is written as f⁻¹(1/5). The function f(x) is given as 2 divided by (x + 1). Alright, let me recall how to find an inverse function.

First, I think the inverse function essentially undoes what the original function does. So if f(x) takes an x and gives me 2/(x + 1), then f⁻¹(y) should take a y and give me back the original x that produced y when plugged into f. So basically, I need to solve the equation y = 2/(x + 1) for x in terms of y, and then substitute y with 1/5.

Let me write that down step by step. Starting with the equation:

y = 2 / (x + 1)

To solve for x, I can start by multiplying both sides of the equation by (x + 1) to get rid of the denominator. Let me do that:

y * (x + 1) = 2

Expanding the left side:

y * x + y * 1 = 2
yx + y = 2

Now, I need to isolate x. Let me subtract y from both sides:

yx = 2 - y

Then, divide both sides by y to solve for x:

x = (2 - y) / y

So, that gives me x in terms of y. Therefore, the inverse function f⁻¹(y) is (2 - y)/y.

Wait, let me check if that makes sense. If I take f(x) = 2/(x + 1), then applying f⁻¹ to f(x) should give me back x. Let me test that. Let me substitute f(x) into f⁻¹:

f⁻¹(f(x)) = f⁻¹(2/(x + 1)) = [2 - (2/(x + 1))]/[2/(x + 1)]

Let me compute that step by step. The numerator is 2 - 2/(x + 1). To combine these terms, I can write 2 as 2(x + 1)/(x + 1):

[2(x + 1) - 2]/(x + 1) divided by [2/(x + 1)]

So numerator becomes [2x + 2 - 2]/(x + 1) = 2x/(x + 1)

Then dividing by 2/(x + 1) is the same as multiplying by (x + 1)/2:

[2x/(x + 1)] * [(x + 1)/2] = x

Perfect! That simplifies to x, so the inverse function is correct.

Therefore, the inverse function is f⁻¹(y) = (2 - y)/y.

Now, I need to find f⁻¹(1/5). That means substituting y = 1/5 into the inverse function:

f⁻¹(1/5) = [2 - (1/5)] / (1/5)

Let me compute the numerator first. 2 - 1/5. Since 2 is 10/5, subtracting 1/5 gives 9/5.

So the numerator is 9/5, and the denominator is 1/5. Dividing these two fractions:

(9/5) divided by (1/5) is equal to (9/5) * (5/1) = 9.

Wait, that's straightforward. So f⁻¹(1/5) is 9? Let me verify that.

Let me check by plugging x = 9 into the original function f(x) and see if I get 1/5.

f(9) = 2 / (9 + 1) = 2/10 = 1/5.

Yes! That's exactly what we wanted. So f(9) = 1/5, which means that f⁻¹(1/5) is indeed 9.

But let me just go through the steps again to make sure I didn't make any mistakes.

Original function: f(x) = 2/(x + 1).

Finding inverse:

Start with y = 2/(x + 1)

Multiply both sides by (x + 1): y(x + 1) = 2

Expand: yx + y = 2

Subtract y: yx = 2 - y

Divide by y: x = (2 - y)/y

Therefore, inverse function f⁻¹(y) = (2 - y)/y.

Then plugging y = 1/5:

(2 - 1/5) / (1/5) = (10/5 - 1/5) / (1/5) = (9/5) / (1/5) = 9.

All steps check out, and verifying by plugging back into original function worked.

Alternatively, another way to think about inverse functions is that if f(a) = b, then f⁻¹(b) = a. So here, we need to find a such that f(a) = 1/5. That is, solve 2/(a + 1) = 1/5.

Cross-multiplying: 2 * 5 = 1 * (a + 1)
10 = a + 1
Therefore, a = 10 - 1 = 9.

Same result. So that's another way to see it.

Therefore, regardless of the method, the answer is 9.

I don't think I made any mistakes here. Both methods of finding the inverse function and then substituting, and directly solving for x when f(x) = 1/5 give me the same answer.

Just to recap, the key steps were:

1. Understanding that finding f⁻¹(1/5) requires solving f(x) = 1/5 for x.
2. Setting up the equation 2/(x + 1) = 1/5.
3. Solving for x by cross-multiplying and simplifying.
4. Confirming the result by plugging back into the original function.

Alternatively, deriving the inverse function explicitly and then substituting the value. Both approaches lead to the same conclusion.

Therefore, I can be confident that the answer is 9.

**Final Answer**
The value of $ f^{-1}\left(\frac{1}{5}\right) $ is \boxed{9}.

To find the value of $ f^{-1}\left(\frac{1}{5}\right) $, we begin by understanding the nature of the inverse function. Given:

$$
f(x) = \frac{2}{x + 1}
$$

We are looking for the value of the inverse function $ f^{-1} $ at $ \frac{1}{5} $, which is equivalent to finding the input $ x $ such that:

$$
f(x) = \frac{1}{5}
$$

---

### Step 1: Set the function equal to $ \frac{1}{5} $

$$
\frac{2}{x + 1} = \frac{1}{5}
$$

---

### Step 2: Solve for $ x $

Start by cross-multiplying:

$$
2 \cdot 5 = (x + 1) \cdot 1
$$

$$
10 = x + 1
$$

$$
x = 9
$$

---

### Step 3: Verify the solution

Plug $ x = 9 $ into the original function:

$$
f(9) = \frac{2}{9 + 1} = \frac{2}{10} = \frac{1}{5}
$$

The result checks out, confirming that $ f(9) = \frac{1}{5} $, so $ f^{-1} \left( \frac{1}{5} \right) = 9 $.

---

### Final Answer

$$
\boxed{9}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of the inverse function of f at 1/5, which is written as f⁻¹(1/5). The function f(x) is given as 2 divided by (x + 1). Alright, let me recall how to find an inverse function.

First, I think the inverse function essentially undoes what the original function does. So if f(x) takes an x and gives me 2/(x + 1), then f⁻¹(y) should take a y and give me back the original x that produced y when plugged into f. So basically, I need to solve the equation y = 2/(x + 1) for x in terms of y, and then substitute y with 1/5.

Let me write that down step by step. Starting with the equation:

y = 2 / (x + 1)

To solve for x, I can start by multiplying both sides of the equation by (x + 1) to get rid of the denominator. Let me do that:

y * (x + 1) = 2

Expanding the left side:

y * x + y * 1 = 2
yx + y = 2

Now, I need to isolate x. Let me subtract y from both sides:

yx = 2 - y

Then, divide both sides by y to solve for x:

x = (2 - y) / y

So, that gives me x in terms of y. Therefore, the inverse function f⁻¹(y) is (2 - y)/y.

Wait, let me check if that makes sense. If I take f(x) = 2/(x + 1), then applying f⁻¹ to f(x) should give me back x. Let me test that. Let me substitute f(x) into f⁻¹:

f⁻¹(f(x)) = f⁻¹(2/(x + 1)) = [2 - (2/(x + 1))]/[2/(x + 1)]

Let me compute that step by step. The numerator is 2 - 2/(x + 1). To combine these terms, I can write 2 as 2(x + 1)/(x + 1):

[2(x + 1) - 2]/(x + 1) divided by [2/(x + 1)]

So numerator becomes [2x + 2 - 2]/(x + 1) = 2x/(x + 1)

Then dividing by 2/(x + 1) is the same as multiplying by (x + 1)/2:

[2x/(x + 1)] * [(x + 1)/2] = x

Perfect! That simplifies to x, so the inverse function is correct.

Therefore, the inverse function is f⁻¹(y) = (2 - y)/y.

Now, I need to find f⁻¹(1/5). That means substituting y = 1/5 into the inverse function:

f⁻¹(1/5) = [2 - (1/5)] / (1/5)

Let me compute the numerator first. 2 - 1/5. Since 2 is 10/5, subtracting 1/5 gives 9/5.

So the numerator is 9/5, and the denominator is 1/5. Dividing these two fractions:

(9/5) divided by (1/5) is equal to (9/5) * (5/1) = 9.

Wait, that's straightforward. So f⁻¹(1/5) is 9? Let me verify that.

Let me check by plugging x = 9 into the original function f(x) and see if I get 1/5.

f(9) = 2 / (9 + 1) = 2/10 = 1/5.

Yes! That's exactly what we wanted. So f(9) = 1/5, which means that f⁻¹(1/5) is indeed 9.

But let me just go through the steps again to make sure I didn't make any mistakes.

Original function: f(x) = 2/(x + 1).

Finding inverse:

Start with y = 2/(x + 1)

Multiply both sides by (x + 1): y(x + 1) = 2

Expand: yx + y = 2

Subtract y: yx = 2 - y

Divide by y: x = (2 - y)/y

Therefore, inverse function f⁻¹(y) = (2 - y)/y.

Then plugging y = 1/5:

(2 - 1/5) / (1/5) = (10/5 - 1/5) / (1/5) = (9/5) / (1/5) = 9.

All steps check out, and verifying by plugging back into original function worked.

Alternatively, another way to think about inverse functions is that if f(a) = b, then f⁻¹(b) = a. So here, we need to find a such that f(a) = 1/5. That is, solve 2/(a + 1) = 1/5.

Cross-multiplying: 2 * 5 = 1 * (a + 1)
10 = a + 1
Therefore, a = 10 - 1 = 9.

Same result. So that's another way to see it.

Therefore, regardless of the method, the answer is 9.

I don't think I made any mistakes here. Both methods of finding the inverse function and then substituting, and directly solving for x when f(x) = 1/5 give me the same answer.

Just to recap, the key steps were:

1. Understanding that finding f⁻¹(1/5) requires solving f(x) = 1/5 for x.
2. Setting up the equation 2/(x + 1) = 1/5.
3. Solving for x by cross-multiplying and simplifying.
4. Confirming the result by plugging back into the original function.

Alternatively, deriving the inverse function explicitly and then substituting the value. Both approaches lead to the same conclusion.

Therefore, I can be confident that the answer is 9.

**Final Answer**
The value of $ f^{-1}\left(\frac{1}{5}\right) $ is \boxed{9}.

To find the value of $ f^{-1}\left(\frac{1}{5}\right) $, we begin by understanding the nature of the inverse function. Given:

$$
f(x) = \frac{2}{x + 1}
$$

We are looking for the value of the inverse function $ f^{-1} $ at $ \frac{1}{5} $, which is equivalent to finding the input $ x $ such that:

$$
f(x) = \frac{1}{5}
$$

---

### Step 1: Set the function equal to $ \frac{1}{5} $

$$
\frac{2}{x + 1} = \frac{1}{5}
$$

---

### Step 2: Solve for $ x $

Start by cross-multiplying:

$$
2 \cdot 5 = (x + 1) \cdot 1
$$

$$
10 = x + 1
$$

$$
x = 9
$$

---

### Step 3: Verify the solution

Plug $ x = 9 $ into the original function:

$$
f(9) = \frac{2}{9 + 1} = \frac{2}{10} = \frac{1}{5}
$$

The result checks out, confirming that $ f(9) = \frac{1}{5} $, so $ f^{-1} \left( \frac{1}{5} \right) = 9 $.

---

### Final Answer

$$
\boxed{9}
$$

Solution: $f^{-1}\left(\frac{1}{5}\right)$ is defined as the number $x$ such that $f(x)=\frac{1}{5}$. Thus, we solve the equation $$\frac{2}{x+1} = \frac{1}{5}.$$Multiplying both sides by $5(x+1)$, we have $$10 = x+1.$$Subtracting $1$ from both sides gives $x=\boxed{9}$.


Explanation:
Okay, let me try to figure out how to solve this hyperbola problem. So, the question says that the hyperbola has asymptotes y = 2x - 3 and y = 17 - 2x, and it passes through the point (4,7). We need to find the distance between the foci of the hyperbola. Hmm, okay, let me recall what I know about hyperbolas.

First, hyperbolas have two asymptotes, which are straight lines that the hyperbola approaches but never touches. The equations of the asymptotes depend on the orientation of the hyperbola. If the hyperbola is oriented horizontally, the standard form is ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1, and the asymptotes are y = k ± (b/a)(x - h). If it's oriented vertically, it's ((y - k)^2)/b^2 - ((x - h)^2)/a^2 = 1, with asymptotes y = k ± (b/a)(x - h) as well. Wait, actually, no, maybe I need to check that again. For horizontal hyperbola, the asymptotes are y - k = ±(b/a)(x - h), and for vertical hyperbola, it's y - k = ±(a/b)(x - h)? Wait, actually, maybe I mixed up a and b. Let me check that.

Let me recall: For hyperbola ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1, the slopes of the asymptotes are ±b/a. Similarly, for ((y - k)^2)/b^2 - ((x - h)^2)/a^2 = 1, the slopes are ±b/a as well? Wait, no, that can't be. Wait, actually, if it's vertical, then the standard form is ((y - k)^2)/a^2 - ((x - h)^2)/b^2 = 1, and then the slopes are ±a/b. Wait, now I'm confused. Let me check.

Let me think of the standard form equations. For horizontal hyperbola, centered at (h,k), the equation is [(x - h)^2]/a^2 - [(y - k)^2]/b^2 = 1. The asymptotes are y - k = ±(b/a)(x - h). For vertical hyperbola, equation is [(y - k)^2]/a^2 - [(x - h)^2]/b^2 = 1, and the asymptotes are y - k = ±(a/b)(x - h). Yes, that seems right because the slopes are determined by the ratio of the denominators. So, if the term with y is positive, then the slopes are ±(a/b), and for the horizontal hyperbola, the slopes are ±(b/a). So the slopes depend on the orientation.

Now, in our problem, the asymptotes are given as y = 2x - 3 and y = 17 - 2x. Let me note that these two lines are not perpendicular. Wait, the slopes are 2 and -2, so they are negative reciprocals? No, 2 and -2 are negative of each other, not reciprocals. So their product is -4, which is not -1. So they are not perpendicular. For hyperbola asymptotes, they are usually symmetric with respect to the center of the hyperbola. So the center of the hyperbola is the intersection point of the asymptotes. So maybe I can first find the center by solving the two asymptote equations.

Yes, that seems like a good first step. Let me find the intersection point of the asymptotes. The asymptotes are y = 2x - 3 and y = 17 - 2x. So, set them equal:

2x - 3 = 17 - 2x

Adding 2x to both sides:

4x - 3 = 17

Adding 3 to both sides:

4x = 20

Divide by 4:

x = 5

Then plug back into one of the equations to find y. Let me use y = 2x - 3:

y = 2*(5) - 3 = 10 - 3 = 7

So the center of the hyperbola is at (5, 7). Wait, that's interesting because the hyperbola passes through the point (4, 7). So the center is (5,7), which is just one unit to the right of the point (4,7). Interesting.

So, center at (h,k) is (5,7). Now, the asymptotes have slopes 2 and -2. Since the slopes are ±2, that tells me about the orientation of the hyperbola. Let me check.

If the slopes are ±m, then for horizontal hyperbola, m = b/a, and for vertical hyperbola, m = a/b. Wait, which is which? For example, if the hyperbola is horizontal, the asymptotes have slopes ±(b/a). If it's vertical, slopes are ±(a/b). Since in our case, the slopes are ±2, so if it's horizontal, then b/a = 2, so b = 2a. If vertical, then a/b = 2, so a = 2b. But I need to figure out which case we have here.

But how can I determine the orientation? Well, given that the hyperbola passes through (4,7). Let me think. Alternatively, maybe we can figure out based on the asymptotes. Alternatively, let's think about the equations.

Alternatively, since we know the center and the slopes of the asymptotes, perhaps we can write the equation of the hyperbola in the form:

(y - k) - (slope1)(x - h) = ± something? Wait, no, more systematically.

The standard equation of a hyperbola with center (h,k) and asymptotes y - k = ±m(x - h) is [(y - k)^2]/a^2 - [(x - h)^2]/b^2 = 1 if it's vertical, or [(x - h)^2]/a^2 - [(y - k)^2]/b^2 = 1 if horizontal, with slopes ±(b/a) or ±(a/b) respectively.

But given that the slopes are ±2, let me think. Let me suppose that the hyperbola is horizontal. Then the slopes would be ±(b/a) = ±2, so b/a = 2, meaning b = 2a. Then, the equation would be [(x - 5)^2]/a^2 - [(y - 7)^2]/b^2 = 1, with b = 2a. So substituting, [(x - 5)^2]/a^2 - [(y - 7)^2]/(4a^2) = 1. Alternatively, if it's vertical, then slopes are ±(a/b) = ±2, so a/b = 2, hence a = 2b. Then the equation would be [(y - 7)^2]/a^2 - [(x - 5)^2]/b^2 = 1, with a = 2b. So [(y - 7)^2]/(4b^2) - [(x - 5)^2]/b^2 = 1. Hmm. So depending on the case, different equations. But how do I know which orientation is correct?

Alternatively, since the hyperbola passes through (4,7), which is a point on the hyperbola. Let's check both possibilities.

First, let me note that the center is (5,7), and the point (4,7) is one unit to the left of the center. So, let's see if that point is on the hyperbola. Let me plug into both cases.

First, assume horizontal hyperbola: [(x - 5)^2]/a^2 - [(y - 7)^2]/(4a^2) = 1. Plug in (4,7):

[(4 - 5)^2]/a^2 - [(7 - 7)^2]/(4a^2) = [1/a^2] - [0] = 1/a^2 = 1. Therefore, 1/a^2 = 1 => a^2 = 1 => a = 1. Therefore, b^2 = 4a^2 = 4. Therefore, equation would be [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. So that's a horizontal hyperbola with a^2 = 1, b^2 = 4. Then, the foci are located at (h ± c, k), where c^2 = a^2 + b^2 = 1 + 4 = 5, so c = sqrt(5). Then distance between foci is 2c = 2*sqrt(5). So that would be the answer?

Wait, but let me check if this is correct. Wait, but maybe the hyperbola is vertical. Let's check that case as well. If vertical hyperbola, then [(y - 7)^2]/(4b^2) - [(x - 5)^2]/b^2 = 1. Let me plug in (4,7):

[(7 - 7)^2]/(4b^2) - [(4 - 5)^2]/b^2 = 0 - [1]/b^2 = -1/b^2 = 1. Therefore, -1/b^2 = 1 => 1/b^2 = -1, which is impossible. Therefore, this case is invalid. Therefore, the hyperbola can't be vertical. Therefore, the correct case is horizontal hyperbola. Therefore, the equation is [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1, and the distance between the foci is 2c = 2*sqrt(5). Therefore, the answer would be 2√5. Wait, but let me confirm that.

Wait, but before that, maybe I made a mistake in assuming that the hyperbola is horizontal? Let me think again. Because if the slopes are ±2, then if the hyperbola is horizontal, then slopes are ±(b/a)= ±2, so that's possible. For vertical hyperbola, slopes are ±(a/b)= ±2. But when we tried plugging in the point (4,7) into the vertical case, we ended up with an impossible equation. So that suggests that the hyperbola is horizontal. Therefore, the answer is 2√5? Wait, but let me check again with the point (4,7). Let me confirm that the point (4,7) is indeed on the hyperbola.

So for horizontal hyperbola: [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Plug in x=4, y=7:

( (4 - 5)^2 ) /1 - ( (7 - 7)^2 ) /4 = 1 - 0 = 1, which satisfies the equation. So that works. Therefore, that seems correct. Then, the distance between foci is 2c, where c^2 = a^2 + b^2 = 1 + 4 = 5. Therefore, c = sqrt(5), so distance is 2*sqrt(5). Therefore, answer is 2√5.

Wait, but let me think again. Is there another way to check? For example, maybe the equation of hyperbola can be written in terms of the asymptotes? Like, the equation of hyperbola can be written as (y - k - m1(x - h))(y - k - m2(x - h)) = c, where c is a constant. Since the asymptotes are y - k = m1(x - h) and y - k = m2(x - h), then multiplying them gives the equation of hyperbola (with some constant). Let me try this method, to verify.

So given the asymptotes y = 2x - 3 and y = 17 - 2x. The center is (5,7). So let me write the asymptotes in terms of the center. So, for the first asymptote, y - 7 = 2(x - 5) - 3 + 7? Wait, no. Wait, original asymptotes are y = 2x -3 and y = 17 - 2x. The center is (5,7). Let me express the asymptotes in the form y - k = m(x - h). For the first asymptote, y = 2x - 3. So, y - 7 = 2x - 3 - 7 => y - 7 = 2x - 10. Therefore, y - 7 = 2(x - 5). Similarly, for the second asymptote y = 17 - 2x, so y - 7 = 17 - 2x -7 = 10 - 2x => y - 7 = -2(x - 5). Therefore, the asymptotes are y - 7 = ±2(x - 5). Therefore, the hyperbola can be written as (y - 7 - 2(x - 5))(y - 7 + 2(x - 5)) = c. Let me compute that.

Multiply the two expressions: [y - 7 - 2(x - 5)][y - 7 + 2(x - 5)] = [ (y -7)^2 - [2(x -5)]^2 ] = (y -7)^2 - 4(x - 5)^2 = c. Therefore, the equation of hyperbola is (y -7)^2 - 4(x -5)^2 = c. Now, since the hyperbola passes through (4,7), plug that point into the equation:

Left side: (7 -7)^2 - 4(4 -5)^2 = 0 - 4(1) = -4. Therefore, c = -4. Therefore, equation is (y -7)^2 - 4(x -5)^2 = -4. If we divide both sides by -4, we get [(y -7)^2]/(-4) - [(x -5)^2]/1 = 1. Wait, but that would be [(x -5)^2]/1 - [(y -7)^2]/4 = 1, which is the same as the equation I found earlier. So that confirms that the equation is indeed [(x -5)^2]/1 - [(y -7)^2]/4 = 1, which is a horizontal hyperbola with a^2 = 1, b^2 = 4. Therefore, c^2 = a^2 + b^2 = 1 + 4 = 5. Therefore, c = √5. Therefore, the distance between the foci is 2c = 2√5. So that's the same result as before.

Therefore, the answer should be 2√5. But let me check once more. Let me check the foci coordinates. Since it's a horizontal hyperbola, the foci are located at (h ± c, k). So center is (5,7), so foci are at (5 + √5, 7) and (5 - √5, 7). The distance between them is [5 + √5 - (5 - √5)] = 2√5. Yes, that's correct. So seems like the answer is 2√5.

But just to make sure there isn't another possible hyperbola, maybe I need to check if there's another orientation? But earlier when I tried vertical hyperbola, plugging in the point (4,7) led to a contradiction. So maybe there is only one possible hyperbola. Let me think again.

Alternatively, let me consider the general equation of hyperbola with asymptotes y = 2x - 3 and y = 17 - 2x. As we found the center at (5,7), and the asymptotes are y -7 = ±2(x -5). Therefore, the hyperbola can be represented as (y -7 - 2(x -5))(y -7 + 2(x -5)) = k, where k is a constant. Which simplifies to (y -7)^2 - [2(x -5)]^2 = k. So (y -7)^2 -4(x -5)^2 = k. Then, since the hyperbola passes through (4,7), substituting, we get 0 -4(1)^2 = -4 = k. Therefore, the equation is (y -7)^2 -4(x -5)^2 = -4. If we divide both sides by -4, we get [(x -5)^2]/1 - [(y -7)^2]/4 = 1. So that's a horizontal hyperbola. Therefore, this is unique. There's no other hyperbola with those asymptotes passing through that point. Therefore, the answer is 2√5. Therefore, I think this is correct.

But just to be thorough, let me check if there is another way to represent hyperbola equations. For example, sometimes hyperbola equations can be written as (y - k)^2 / a^2 - (x - h)^2 / b^2 = 1 or vice versa. But in our case, we found that the equation is [(x -5)^2]/1 - [(y -7)^2]/4 = 1, which is horizontal. So that's the only possibility. Therefore, yes, the distance between foci is 2c, which is 2*sqrt(a² + b²) = 2*sqrt(1 + 4) = 2*sqrt(5).

Therefore, I think that the answer is 2√5, so boxed answer would be \boxed{2\sqrt{5}}.

**Final Answer**
\boxed{2\sqrt{5}}

To find the **distance between the foci** of a hyperbola, we begin by leveraging the information provided:

- The **asymptotes** of the hyperbola are:
  $ y = 2x - 3 $ and $ y = 17 - 2x $.
- The hyperbola **passes through** the point $ (4,7) $.

---

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

The **center** of the hyperbola is the point of intersection of its asymptotes. To find this:

$$
2x - 3 = 17 - 2x \Rightarrow 4x = 20 \Rightarrow x = 5
$$
$$
y = 2(5) - 3 = 10 - 3 = 7
$$

So the center of the hyperbola is at $ (5,7) $.

---

### Step 2: Determine the Orientation of the Hyperbola

The asymptotes are:

- $ y = 2x - 3 $: Slope $ = 2 $
- $ y = 17 - 2x $: Slope $ = -2 $

The slopes are $ \pm2 $, which are symmetric about the center. For a hyperbola centered at $ (h,k) $, if its asymptotes are $ y - k = \pm m(x - h) $, then:

- If the slopes are $ \pm m $, then the **horizontal** hyperbola has asymptotes with slopes $ \pm \frac{b}{a} $, and the **vertical** hyperbola has slopes $ \pm \frac{a}{b} $.

Since the product of the slopes is $ -4 $, it's not perpendicular, so it's not a rectangular hyperbola, but still matches either a horizontal or vertical hyperbola.

We'll consider both possibilities and check which one is consistent with the point $ (4,7) $.

---

### Step 3: Assume the Hyperbola is Horizontal

Let the equation of the hyperbola be:

$$
\frac{(x - 5)^2}{a^2} - \frac{(y - 7)^2}{b^2} = 1
$$

The asymptotes are:

$$
y - 7 = \pm \frac{b}{a}(x - 5)
$$

So the slopes are $ \pm \frac{b}{a} = \pm 2 \Rightarrow \frac{b}{a} = 2 \Rightarrow b = 2a $

Now plug in the point $ (4,7) $ into the equation:

$$
\frac{(4 - 5)^2}{a^2} - \frac{(7 - 7)^2}{b^2} = 1 \Rightarrow \frac{1}{a^2} = 1 \Rightarrow a^2 = 1 \Rightarrow a = 1
$$

Then $ b^2 = (2a)^2 = 4 $

So the equation is:

$$
\frac{(x - 5)^2}{1} - \frac{(y - 7)^2}{4} = 1
$$

Which is a valid horizontal hyperbola.

---

### Step 4: Find the Distance Between the Foci

For a horizontal hyperbola:

- The distance between the foci is $ 2c $, where:
  $$
  c^2 = a^2 + b^2 = 1 + 4 = 5 \Rightarrow c = \sqrt{5}
  $$
- So the distance between the foci is:
  $$
  2c = 2\sqrt{5}
  $$

---

### Step 5: Verification via Asymptotes

The asymptotes are $ y - 7 = \pm 2(x - 5) $, which correspond exactly to our derived slopes. Furthermore, the standard form with this center, and slopes $ \pm 2 $, implies that the hyperbola must be horizontal. Any vertical assumption leads to inconsistency when plugging in the point $ (4,7) $.

---

### ✅ Final Answer

$$
\boxed{2\sqrt{5}}
$$

[Assistant]
Explanation:
Okay, let me try to figure out how to solve this hyperbola problem. So, the question says that the hyperbola has asymptotes y = 2x - 3 and y = 17 - 2x, and it passes through the point (4,7). We need to find the distance between the foci of the hyperbola. Hmm, okay, let me recall what I know about hyperbolas.

First, hyperbolas have two asymptotes, which are straight lines that the hyperbola approaches but never touches. The equations of the asymptotes depend on the orientation of the hyperbola. If the hyperbola is oriented horizontally, the standard form is ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1, and the asymptotes are y = k ± (b/a)(x - h). If it's oriented vertically, it's ((y - k)^2)/b^2 - ((x - h)^2)/a^2 = 1, with asymptotes y = k ± (b/a)(x - h) as well. Wait, actually, no, maybe I need to check that again. For horizontal hyperbola, the asymptotes are y - k = ±(b/a)(x - h), and for vertical hyperbola, it's y - k = ±(a/b)(x - h)? Wait, actually, maybe I mixed up a and b. Let me check that.

Let me recall: For hyperbola ((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1, the slopes of the asymptotes are ±b/a. Similarly, for ((y - k)^2)/b^2 - ((x - h)^2)/a^2 = 1, the slopes are ±b/a as well? Wait, no, that can't be. Wait, actually, if it's vertical, then the standard form is ((y - k)^2)/a^2 - ((x - h)^2)/b^2 = 1, and then the slopes are ±a/b. Wait, now I'm confused. Let me check.

Let me think of the standard form equations. For horizontal hyperbola, centered at (h,k), the equation is [(x - h)^2]/a^2 - [(y - k)^2]/b^2 = 1. The asymptotes are y - k = ±(b/a)(x - h). For vertical hyperbola, equation is [(y - k)^2]/a^2 - [(x - h)^2]/b^2 = 1, and the asymptotes are y - k = ±(a/b)(x - h). Yes, that seems right because the slopes are determined by the ratio of the denominators. So, if the term with y is positive, then the slopes are ±(a/b), and for the horizontal hyperbola, the slopes are ±(b/a). So the slopes depend on the orientation.

Now, in our problem, the asymptotes are given as y = 2x - 3 and y = 17 - 2x. Let me note that these two lines are not perpendicular. Wait, the slopes are 2 and -2, so they are negative reciprocals? No, 2 and -2 are negative of each other, not reciprocals. So their product is -4, which is not -1. So they are not perpendicular. For hyperbola asymptotes, they are usually symmetric with respect to the center of the hyperbola. So the center of the hyperbola is the intersection point of the asymptotes. So maybe I can first find the center by solving the two asymptote equations.

Yes, that seems like a good first step. Let me find the intersection point of the asymptotes. The asymptotes are y = 2x - 3 and y = 17 - 2x. So, set them equal:

2x - 3 = 17 - 2x

Adding 2x to both sides:

4x - 3 = 17

Adding 3 to both sides:

4x = 20

Divide by 4:

x = 5

Then plug back into one of the equations to find y. Let me use y = 2x - 3:

y = 2*(5) - 3 = 10 - 3 = 7

So the center of the hyperbola is at (5, 7). Wait, that's interesting because the hyperbola passes through the point (4, 7). So the center is (5,7), which is just one unit to the right of the point (4,7). Interesting.

So, center at (h,k) is (5,7). Now, the asymptotes have slopes 2 and -2. Since the slopes are ±2, that tells me about the orientation of the hyperbola. Let me check.

If the slopes are ±m, then for horizontal hyperbola, m = b/a, and for vertical hyperbola, m = a/b. Wait, which is which? For example, if the hyperbola is horizontal, the asymptotes have slopes ±(b/a). If it's vertical, slopes are ±(a/b). Since in our case, the slopes are ±2, so if it's horizontal, then b/a = 2, so b = 2a. If vertical, then a/b = 2, so a = 2b. But I need to figure out which case we have here.

But how can I determine the orientation? Well, given that the hyperbola passes through (4,7). Let me think. Alternatively, maybe we can figure out based on the asymptotes. Alternatively, let's think about the equations.

Alternatively, since we know the center and the slopes of the asymptotes, perhaps we can write the equation of the hyperbola in the form:

(y - k) - (slope1)(x - h) = ± something? Wait, no, more systematically.

The standard equation of a hyperbola with center (h,k) and asymptotes y - k = ±m(x - h) is [(y - k)^2]/a^2 - [(x - h)^2]/b^2 = 1 if it's vertical, or [(x - h)^2]/a^2 - [(y - k)^2]/b^2 = 1 if horizontal, with slopes ±(b/a) or ±(a/b) respectively.

But given that the slopes are ±2, let me think. Let me suppose that the hyperbola is horizontal. Then the slopes would be ±(b/a) = ±2, so b/a = 2, meaning b = 2a. Then, the equation would be [(x - 5)^2]/a^2 - [(y - 7)^2]/b^2 = 1, with b = 2a. So substituting, [(x - 5)^2]/a^2 - [(y - 7)^2]/(4a^2) = 1. Alternatively, if it's vertical, then slopes are ±(a/b) = ±2, so a/b = 2, hence a = 2b. Then the equation would be [(y - 7)^2]/a^2 - [(x - 5)^2]/b^2 = 1, with a = 2b. So [(y - 7)^2]/(4b^2) - [(x - 5)^2]/b^2 = 1. Hmm. So depending on the case, different equations. But how do I know which orientation is correct?

Alternatively, since the hyperbola passes through (4,7), which is a point on the hyperbola. Let's check both possibilities.

First, let me note that the center is (5,7), and the point (4,7) is one unit to the left of the center. So, let's see if that point is on the hyperbola. Let me plug into both cases.

First, assume horizontal hyperbola: [(x - 5)^2]/a^2 - [(y - 7)^2]/(4a^2) = 1. Plug in (4,7):

[(4 - 5)^2]/a^2 - [(7 - 7)^2]/(4a^2) = [1/a^2] - [0] = 1/a^2 = 1. Therefore, 1/a^2 = 1 => a^2 = 1 => a = 1. Therefore, b^2 = 4a^2 = 4. Therefore, equation would be [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. So that's a horizontal hyperbola with a^2 = 1, b^2 = 4. Then, the foci are located at (h ± c, k), where c^2 = a^2 + b^2 = 1 + 4 = 5, so c = sqrt(5). Then distance between foci is 2c = 2*sqrt(5). So that would be the answer?

Wait, but let me check if this is correct. Wait, but maybe the hyperbola is vertical. Let's check that case as well. If vertical hyperbola, then [(y - 7)^2]/(4b^2) - [(x - 5)^2]/b^2 = 1. Let me plug in (4,7):

[(7 - 7)^2]/(4b^2) - [(4 - 5)^2]/b^2 = 0 - [1]/b^2 = -1/b^2 = 1. Therefore, -1/b^2 = 1 => 1/b^2 = -1, which is impossible. Therefore, this case is invalid. Therefore, the hyperbola can't be vertical. Therefore, the correct case is horizontal hyperbola. Therefore, the equation is [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1, and the distance between the foci is 2c = 2*sqrt(5). Therefore, the answer would be 2√5. Wait, but let me confirm that.

Wait, but before that, maybe I made a mistake in assuming that the hyperbola is horizontal? Let me think again. Because if the slopes are ±2, then if the hyperbola is horizontal, then slopes are ±(b/a)= ±2, so that's possible. For vertical hyperbola, slopes are ±(a/b)= ±2. But when we tried plugging in the point (4,7) into the vertical case, we ended up with an impossible equation. So that suggests that the hyperbola is horizontal. Therefore, the answer is 2√5? Wait, but let me check again with the point (4,7). Let me confirm that the point (4,7) is indeed on the hyperbola.

So for horizontal hyperbola: [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Plug in x=4, y=7:

( (4 - 5)^2 ) /1 - ( (7 - 7)^2 ) /4 = 1 - 0 = 1, which satisfies the equation. So that works. Therefore, that seems correct. Then, the distance between foci is 2c, where c^2 = a^2 + b^2 = 1 + 4 = 5. Therefore, c = sqrt(5), so distance is 2*sqrt(5). Therefore, answer is 2√5.

Wait, but let me think again. Is there another way to check? For example, maybe the equation of hyperbola can be written in terms of the asymptotes? Like, the equation of hyperbola can be written as (y - k - m1(x - h))(y - k - m2(x - h)) = c, where c is a constant. Since the asymptotes are y - k = m1(x - h) and y - k = m2(x - h), then multiplying them gives the equation of hyperbola (with some constant). Let me try this method, to verify.

So given the asymptotes y = 2x - 3 and y = 17 - 2x. The center is (5,7). So let me write the asymptotes in terms of the center. So, for the first asymptote, y - 7 = 2(x - 5) - 3 + 7? Wait, no. Wait, original asymptotes are y = 2x -3 and y = 17 - 2x. The center is (5,7). Let me express the asymptotes in the form y - k = m(x - h). For the first asymptote, y = 2x - 3. So, y - 7 = 2x - 3 - 7 => y - 7 = 2x - 10. Therefore, y - 7 = 2(x - 5). Similarly, for the second asymptote y = 17 - 2x, so y - 7 = 17 - 2x -7 = 10 - 2x => y - 7 = -2(x - 5). Therefore, the asymptotes are y - 7 = ±2(x - 5). Therefore, the hyperbola can be written as (y - 7 - 2(x - 5))(y - 7 + 2(x - 5)) = c. Let me compute that.

Multiply the two expressions: [y - 7 - 2(x - 5)][y - 7 + 2(x - 5)] = [ (y -7)^2 - [2(x -5)]^2 ] = (y -7)^2 - 4(x - 5)^2 = c. Therefore, the equation of hyperbola is (y -7)^2 - 4(x -5)^2 = c. Now, since the hyperbola passes through (4,7), plug that point into the equation:

Left side: (7 -7)^2 - 4(4 -5)^2 = 0 - 4(1) = -4. Therefore, c = -4. Therefore, equation is (y -7)^2 - 4(x -5)^2 = -4. If we divide both sides by -4, we get [(y -7)^2]/(-4) - [(x -5)^2]/1 = 1. Wait, but that would be [(x -5)^2]/1 - [(y -7)^2]/4 = 1, which is the same as the equation I found earlier. So that confirms that the equation is indeed [(x -5)^2]/1 - [(y -7)^2]/4 = 1, which is a horizontal hyperbola with a^2 = 1, b^2 = 4. Therefore, c^2 = a^2 + b^2 = 1 + 4 = 5. Therefore, c = √5. Therefore, the distance between the foci is 2c = 2√5. So that's the same result as before.

Therefore, the answer should be 2√5. But let me check once more. Let me check the foci coordinates. Since it's a horizontal hyperbola, the foci are located at (h ± c, k). So center is (5,7), so foci are at (5 + √5, 7) and (5 - √5, 7). The distance between them is [5 + √5 - (5 - √5)] = 2√5. Yes, that's correct. So seems like the answer is 2√5.

But just to make sure there isn't another possible hyperbola, maybe I need to check if there's another orientation? But earlier when I tried vertical hyperbola, plugging in the point (4,7) led to a contradiction. So maybe there is only one possible hyperbola. Let me think again.

Alternatively, let me consider the general equation of hyperbola with asymptotes y = 2x - 3 and y = 17 - 2x. As we found the center at (5,7), and the asymptotes are y -7 = ±2(x -5). Therefore, the hyperbola can be represented as (y -7 - 2(x -5))(y -7 + 2(x -5)) = k, where k is a constant. Which simplifies to (y -7)^2 - [2(x -5)]^2 = k. So (y -7)^2 -4(x -5)^2 = k. Then, since the hyperbola passes through (4,7), substituting, we get 0 -4(1)^2 = -4 = k. Therefore, the equation is (y -7)^2 -4(x -5)^2 = -4. If we divide both sides by -4, we get [(x -5)^2]/1 - [(y -7)^2]/4 = 1. So that's a horizontal hyperbola. Therefore, this is unique. There's no other hyperbola with those asymptotes passing through that point. Therefore, the answer is 2√5. Therefore, I think this is correct.

But just to be thorough, let me check if there is another way to represent hyperbola equations. For example, sometimes hyperbola equations can be written as (y - k)^2 / a^2 - (x - h)^2 / b^2 = 1 or vice versa. But in our case, we found that the equation is [(x -5)^2]/1 - [(y -7)^2]/4 = 1, which is horizontal. So that's the only possibility. Therefore, yes, the distance between foci is 2c, which is 2*sqrt(a² + b²) = 2*sqrt(1 + 4) = 2*sqrt(5).

Therefore, I think that the answer is 2√5, so boxed answer would be \boxed{2\sqrt{5}}.

**Final Answer**
\boxed{2\sqrt{5}}

To find the **distance between the foci** of a hyperbola, we begin by leveraging the information provided:

- The **asymptotes** of the hyperbola are:
  $ y = 2x - 3 $ and $ y = 17 - 2x $.
- The hyperbola **passes through** the point $ (4,7) $.

---

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

The **center** of the hyperbola is the point of intersection of its asymptotes. To find this:

$$
2x - 3 = 17 - 2x \Rightarrow 4x = 20 \Rightarrow x = 5
$$
$$
y = 2(5) - 3 = 10 - 3 = 7
$$

So the center of the hyperbola is at $ (5,7) $.

---

### Step 2: Determine the Orientation of the Hyperbola

The asymptotes are:

- $ y = 2x - 3 $: Slope $ = 2 $
- $ y = 17 - 2x $: Slope $ = -2 $

The slopes are $ \pm2 $, which are symmetric about the center. For a hyperbola centered at $ (h,k) $, if its asymptotes are $ y - k = \pm m(x - h) $, then:

- If the slopes are $ \pm m $, then the **horizontal** hyperbola has asymptotes with slopes $ \pm \frac{b}{a} $, and the **vertical** hyperbola has slopes $ \pm \frac{a}{b} $.

Since the product of the slopes is $ -4 $, it's not perpendicular, so it's not a rectangular hyperbola, but still matches either a horizontal or vertical hyperbola.

We'll consider both possibilities and check which one is consistent with the point $ (4,7) $.

---

### Step 3: Assume the Hyperbola is Horizontal

Let the equation of the hyperbola be:

$$
\frac{(x - 5)^2}{a^2} - \frac{(y - 7)^2}{b^2} = 1
$$

The asymptotes are:

$$
y - 7 = \pm \frac{b}{a}(x - 5)
$$

So the slopes are $ \pm \frac{b}{a} = \pm 2 \Rightarrow \frac{b}{a} = 2 \Rightarrow b = 2a $

Now plug in the point $ (4,7) $ into the equation:

$$
\frac{(4 - 5)^2}{a^2} - \frac{(7 - 7)^2}{b^2} = 1 \Rightarrow \frac{1}{a^2} = 1 \Rightarrow a^2 = 1 \Rightarrow a = 1
$$

Then $ b^2 = (2a)^2 = 4 $

So the equation is:

$$
\frac{(x - 5)^2}{1} - \frac{(y - 7)^2}{4} = 1
$$

Which is a valid horizontal hyperbola.

---

### Step 4: Find the Distance Between the Foci

For a horizontal hyperbola:

- The distance between the foci is $ 2c $, where:
  $$
  c^2 = a^2 + b^2 = 1 + 4 = 5 \Rightarrow c = \sqrt{5}
  $$
- So the distance between the foci is:
  $$
  2c = 2\sqrt{5}
  $$

---

### Step 5: Verification via Asymptotes

The asymptotes are $ y - 7 = \pm 2(x - 5) $, which correspond exactly to our derived slopes. Furthermore, the standard form with this center, and slopes $ \pm 2 $, implies that the hyperbola must be horizontal. Any vertical assumption leads to inconsistency when plugging in the point $ (4,7) $.

---

### ✅ Final Answer

$$
\boxed{2\sqrt{5}}
$$

Solution:
The intersection of the asymptotes is $(5,7),$ so this is the center of the hyperbola.  Since the slopes of the asymptotes are $\pm 2,$ the equation of the hyperbola can be written in the form
\[(x - 5)^2 - \frac{(y - 7)^2}{4} = d\]for some constant $d.$  Setting $x = 4$ and $y = 7,$ we get $d = 1,$ so the equation is
\[\frac{(x - 5)^2}{1} - \frac{(y - 7)^2}{4} = 1.\]Then $a^2 = 1$ and $b^2 = 4,$ so $c^2 = a^2 + b^2 = 5,$ which means $c = \sqrt{5}.$  Therefore, the distance between the foci is $2c = \boxed{2 \sqrt{5}}.$