Explanation:
Okay, so I need to find the greatest common divisor (GCD) of three numbers: 3339, 2961, and 1491. Hmm, GCD of three numbers... I remember that the GCD of multiple numbers can be found by first finding the GCD of two numbers and then finding the GCD of that result with the third number. So maybe I should start by finding the GCD of 3339 and 2961 first, and then take that result and find the GCD with 1491. Let me try that approach.

First, let me recall the Euclidean algorithm for finding the GCD of two numbers. The Euclidean algorithm involves dividing the larger number by the smaller one, taking the remainder, and then repeating the process with the smaller number and the remainder until the remainder is zero. The last non-zero remainder is the GCD. Okay, so let me apply that to 3339 and 2961.

Let me write down the steps. Let me call the numbers a and b, where a = 3339 and b = 2961. Since a is larger than b, I can start by dividing a by b.

So, 3339 divided by 2961. Let me compute how many times 2961 goes into 3339. Well, 2961 times 1 is 2961, and 2961 times 2 is 5922, which is larger than 3339. So it goes in once with a remainder. The remainder is 3339 - 2961 = 378. So, according to the Euclidean algorithm, GCD(3339, 2961) = GCD(2961, 378).

Now I need to find GCD(2961, 378). Again, divide the larger number by the smaller one. Let me do 2961 divided by 378. Let me see how many times 378 fits into 2961.

First, 378 times 7 is 2646. Then 378 times 8 is 3024, which is larger than 2961. So it fits 7 times, with a remainder. Let me compute the remainder: 2961 - 2646 = 315. Therefore, GCD(2961, 378) = GCD(378, 315).

Now, moving on to GCD(378, 315). Again, divide the larger by the smaller. 378 divided by 315. 315 goes into 378 once, with a remainder of 378 - 315 = 63. Therefore, GCD(378, 315) = GCD(315, 63).

Now, GCD(315, 63). Let me divide 315 by 63. 63 times 5 is 315, so there's no remainder. Therefore, the remainder is zero, which means that the GCD is 63.

So, going back, GCD(3339, 2961) is 63. Now I need to find the GCD of this result (63) with the third number, which is 1491. Therefore, I need to compute GCD(63, 1491).

Again, applying the Euclidean algorithm. Let me note that 1491 is larger than 63, so I can divide 1491 by 63. Let me compute how many times 63 fits into 1491.

First, let me calculate 63 times 20 is 1260. Then, 1491 - 1260 = 231. Then 63 times 3 is 189. So 231 - 189 = 42. Then 63 goes into 42 zero times, so the remainder is 42. Wait, let me check that again. Maybe I should do this more systematically.

Alternatively, divide 1491 by 63. Let me do the division step by step.

First, 63 × 20 = 1260. Subtract that from 1491: 1491 - 1260 = 231. Now divide 231 by 63. 63 × 3 = 189. Subtract that from 231: 231 - 189 = 42. So the remainder is 42. Therefore, GCD(1491, 63) = GCD(63, 42).

Now, compute GCD(63, 42). Divide 63 by 42. 42 goes into 63 once, with a remainder of 63 - 42 = 21. Therefore, GCD(63, 42) = GCD(42, 21).

Now, divide 42 by 21. 21 × 2 = 42, so the remainder is zero. Therefore, the GCD is 21.

So, putting it all together, the GCD of 63 and 1491 is 21. Therefore, the GCD of all three numbers, 3339, 2961, and 1491, is 21.

But wait, let me verify this because it's easy to make mistakes in the calculations. Let me check if 21 divides all three numbers.

First, 3339 divided by 21. Let me compute 21 × 160 = 3360, which is more than 3339. So 21 × 159 = 21 × 160 - 21 = 3360 - 21 = 3339. Perfect, so 21 divides 3339 exactly 159 times.

Next, 2961 divided by 21. Let me compute 21 × 100 = 2100. 2961 - 2100 = 861. 21 × 40 = 840. 861 - 840 = 21. So that's 21 × 1. Therefore, total is 100 + 40 + 1 = 141. Therefore, 21 × 141 = 2961. So yes, 21 divides 2961.

Now check 1491 divided by 21. Let me compute 21 × 70 = 1470. Then 1491 - 1470 = 21. So that's 21 × 1. Therefore, total is 70 + 1 = 71. Therefore, 21 × 71 = 1491. Perfect, so 21 divides all three numbers.

Now, is there a larger number that divides all three? Well, since we found the GCD step by step using the Euclidean algorithm, which is a reliable method, and the result is 21, and we checked that 21 divides all three numbers, and since the algorithm ensures that it's the greatest common divisor, I think that's correct.

But just to make sure, let me try another approach. Maybe prime factorization? Let's see.

First, let me factorize each number.

Starting with 3339. Let me check divisibility by small primes.

3339: sum of digits is 3 + 3 + 3 + 9 = 18, which is divisible by 9, so 3339 is divisible by 9. Let me check: 3339 ÷ 9. 9 × 371 = 3339? Let me compute 9 × 300 = 2700, 9 × 70 = 630, 9 × 1 = 9. So 2700 + 630 = 3330, plus 9 is 3339. So 3339 = 9 × 371. Now factorizing 371. Let me check divisibility. 371 ÷ 7 = 53? 7 × 53 is 371? 7 × 50 = 350, 7 × 3 = 21, so 350 +21 = 371. Yes! So 371 = 7 × 53. Therefore, 3339 = 9 × 7 × 53. Which is 3² × 7 × 53.

Now, 2961. Let me factorize 2961. Again, check divisibility by 3: sum of digits is 2 + 9 + 6 + 1 = 18, which is divisible by 9. Wait, 2961 ÷ 9. Let me compute 9 × 329 = 2961? Let me check: 9 × 300 = 2700, 9 × 29 = 261. 2700 + 261 = 2961. So yes. So 2961 = 9 × 329. Now factorize 329. Let me check divisibility by 7: 7 × 47 = 329? 7 × 40 = 280, 7 × 7 = 49. 280 + 49 = 329. Yes. So 329 = 7 × 47. Therefore, 2961 = 9 × 7 × 47 = 3² × 7 × 47.

Now, factorizing 1491. Let me check divisibility by 3: 1 + 4 + 9 + 1 = 15, which is divisible by 3. 1491 ÷ 3 = 497. Let me check that: 3 × 497 = 1491. Now, factorize 497. Let me check divisibility by 7: 7 × 71 = 497? 7 × 70 = 490, plus 7 is 497. Yes! Therefore, 497 = 7 × 71. Therefore, 1491 = 3 × 7 × 71.

So, putting all together:

- 3339: 3² × 7 × 53
- 2961: 3² × 7 × 47
- 1491: 3 × 7 × 71

Now, the GCD is the product of the lowest power of common primes. Let's see. The common primes in all three numbers are 3 and 7.

For prime 3: in 3339 it's 3², in 2961 it's 3², and in 1491 it's 3¹. Therefore, the minimum exponent is 1.

For prime 7: all three numbers have 7¹. So, the minimum exponent is 1.

Therefore, GCD is 3¹ × 7¹ = 3 × 7 = 21.

So that confirms the earlier result. Therefore, 21 is indeed the greatest common divisor.

Wait, just to be thorough, let me check if there's a common divisor larger than 21. For example, could there be a common divisor of 21 × something? Let's see. Let me check if 21 × 2 = 42 divides all three numbers.

Check 3339 ÷ 42. 42 × 80 = 3360, which is higher than 3339. 42 × 79 = 42 × 80 - 42 = 3360 - 42 = 3318. 3339 - 3318 = 21. So remainder 21, so no. Therefore, 42 does not divide 3339.

Similarly, check 21 × 3 = 63. Let me check 3339 ÷ 63. Earlier, when we were computing GCD(3339, 2961), we found that GCD was 63, but after taking GCD with 1491, it became 21. Wait, but perhaps I need to check if 63 divides 1491? Let's see. 1491 ÷ 63. 63 × 23 = 1449. 1491 - 1449 = 42. So remainder 42, which means 63 does not divide 1491. Therefore, 63 is not a common divisor. Therefore, 21 is indeed the highest.

Alternatively, let me check if there is another common divisor. For example, let's see if there is a common divisor of 3, 7, or 21. Since we have already found that 21 is the GCD, and since 21 is the product of 3 and 7, and the prime factors are as above, there's no higher common factor.

So, seems like 21 is the correct answer.

Just to recap the steps:

1. Found GCD(3339, 2961) using Euclidean algorithm: 63.
2. Then found GCD(63, 1491) using Euclidean algorithm: 21.
3. Verified by prime factorization that the common factors are 3 and 7, so GCD is 3×7=21.
4. Checked that 21 divides all three numbers and that no higher number does.

Therefore, I'm confident that the greatest common divisor is 21.

**Final Answer**
The greatest common divisor of $3339$, $2961$, and $1491$ is \boxed{21}.

To find the **greatest common divisor (GCD)** of the numbers $ 3339 $, $ 2961 $, and $ 1491 $, we apply the **Euclidean algorithm** step by step. The Euclidean algorithm allows us to find the GCD of two numbers by repeatedly applying the remainder operation. If we want the GCD of three numbers, we first find the GCD of two of them and then find the GCD of that result with the third number.

---

### Step 1: Compute GCD(3339, 2961)

Use the Euclidean algorithm:

$$
\begin{align*}
3339 \div 2961 &= 1 \text{ with remainder } 378 \\
2961 \div 378 &= 7 \text{ with remainder } 315 \\
378 \div 315 &= 1 \text{ with remainder } 63 \\
315 \div 63 &= 5 \text{ with remainder } 0 \\
\end{align*}
$$

So, $ \gcd(3339, 2961) = 63 $

---

### Step 2: Compute GCD(63, 1491)

$$
\begin{align*}
1491 \div 63 &= 23 \text{ with remainder } 42 \\
63 \div 42 &= 1 \text{ with remainder } 21 \\
42 \div 21 &= 2 \text{ with remainder } 0 \\
\end{align*}
$$

So, $ \gcd(63, 1491) = 21 $

---

### Step 3: Interpret the Result

Since we now have:

$$
\gcd(3339, 2961, 1491) = \gcd(\gcd(3339, 2961), 1491) = \gcd(63, 1491) = 21
$$

We conclude that the **greatest common divisor** of the three numbers is:

$$
\boxed{21}
$$

[Assistant]
Explanation:
Okay, so I need to find the greatest common divisor (GCD) of three numbers: 3339, 2961, and 1491. Hmm, GCD of three numbers... I remember that the GCD of multiple numbers can be found by first finding the GCD of two numbers and then finding the GCD of that result with the third number. So maybe I should start by finding the GCD of 3339 and 2961 first, and then take that result and find the GCD with 1491. Let me try that approach.

First, let me recall the Euclidean algorithm for finding the GCD of two numbers. The Euclidean algorithm involves dividing the larger number by the smaller one, taking the remainder, and then repeating the process with the smaller number and the remainder until the remainder is zero. The last non-zero remainder is the GCD. Okay, so let me apply that to 3339 and 2961.

Let me write down the steps. Let me call the numbers a and b, where a = 3339 and b = 2961. Since a is larger than b, I can start by dividing a by b.

So, 3339 divided by 2961. Let me compute how many times 2961 goes into 3339. Well, 2961 times 1 is 2961, and 2961 times 2 is 5922, which is larger than 3339. So it goes in once with a remainder. The remainder is 3339 - 2961 = 378. So, according to the Euclidean algorithm, GCD(3339, 2961) = GCD(2961, 378).

Now I need to find GCD(2961, 378). Again, divide the larger number by the smaller one. Let me do 2961 divided by 378. Let me see how many times 378 fits into 2961.

First, 378 times 7 is 2646. Then 378 times 8 is 3024, which is larger than 2961. So it fits 7 times, with a remainder. Let me compute the remainder: 2961 - 2646 = 315. Therefore, GCD(2961, 378) = GCD(378, 315).

Now, moving on to GCD(378, 315). Again, divide the larger by the smaller. 378 divided by 315. 315 goes into 378 once, with a remainder of 378 - 315 = 63. Therefore, GCD(378, 315) = GCD(315, 63).

Now, GCD(315, 63). Let me divide 315 by 63. 63 times 5 is 315, so there's no remainder. Therefore, the remainder is zero, which means that the GCD is 63.

So, going back, GCD(3339, 2961) is 63. Now I need to find the GCD of this result (63) with the third number, which is 1491. Therefore, I need to compute GCD(63, 1491).

Again, applying the Euclidean algorithm. Let me note that 1491 is larger than 63, so I can divide 1491 by 63. Let me compute how many times 63 fits into 1491.

First, let me calculate 63 times 20 is 1260. Then, 1491 - 1260 = 231. Then 63 times 3 is 189. So 231 - 189 = 42. Then 63 goes into 42 zero times, so the remainder is 42. Wait, let me check that again. Maybe I should do this more systematically.

Alternatively, divide 1491 by 63. Let me do the division step by step.

First, 63 × 20 = 1260. Subtract that from 1491: 1491 - 1260 = 231. Now divide 231 by 63. 63 × 3 = 189. Subtract that from 231: 231 - 189 = 42. So the remainder is 42. Therefore, GCD(1491, 63) = GCD(63, 42).

Now, compute GCD(63, 42). Divide 63 by 42. 42 goes into 63 once, with a remainder of 63 - 42 = 21. Therefore, GCD(63, 42) = GCD(42, 21).

Now, divide 42 by 21. 21 × 2 = 42, so the remainder is zero. Therefore, the GCD is 21.

So, putting it all together, the GCD of 63 and 1491 is 21. Therefore, the GCD of all three numbers, 3339, 2961, and 1491, is 21.

But wait, let me verify this because it's easy to make mistakes in the calculations. Let me check if 21 divides all three numbers.

First, 3339 divided by 21. Let me compute 21 × 160 = 3360, which is more than 3339. So 21 × 159 = 21 × 160 - 21 = 3360 - 21 = 3339. Perfect, so 21 divides 3339 exactly 159 times.

Next, 2961 divided by 21. Let me compute 21 × 100 = 2100. 2961 - 2100 = 861. 21 × 40 = 840. 861 - 840 = 21. So that's 21 × 1. Therefore, total is 100 + 40 + 1 = 141. Therefore, 21 × 141 = 2961. So yes, 21 divides 2961.

Now check 1491 divided by 21. Let me compute 21 × 70 = 1470. Then 1491 - 1470 = 21. So that's 21 × 1. Therefore, total is 70 + 1 = 71. Therefore, 21 × 71 = 1491. Perfect, so 21 divides all three numbers.

Now, is there a larger number that divides all three? Well, since we found the GCD step by step using the Euclidean algorithm, which is a reliable method, and the result is 21, and we checked that 21 divides all three numbers, and since the algorithm ensures that it's the greatest common divisor, I think that's correct.

But just to make sure, let me try another approach. Maybe prime factorization? Let's see.

First, let me factorize each number.

Starting with 3339. Let me check divisibility by small primes.

3339: sum of digits is 3 + 3 + 3 + 9 = 18, which is divisible by 9, so 3339 is divisible by 9. Let me check: 3339 ÷ 9. 9 × 371 = 3339? Let me compute 9 × 300 = 2700, 9 × 70 = 630, 9 × 1 = 9. So 2700 + 630 = 3330, plus 9 is 3339. So 3339 = 9 × 371. Now factorizing 371. Let me check divisibility. 371 ÷ 7 = 53? 7 × 53 is 371? 7 × 50 = 350, 7 × 3 = 21, so 350 +21 = 371. Yes! So 371 = 7 × 53. Therefore, 3339 = 9 × 7 × 53. Which is 3² × 7 × 53.

Now, 2961. Let me factorize 2961. Again, check divisibility by 3: sum of digits is 2 + 9 + 6 + 1 = 18, which is divisible by 9. Wait, 2961 ÷ 9. Let me compute 9 × 329 = 2961? Let me check: 9 × 300 = 2700, 9 × 29 = 261. 2700 + 261 = 2961. So yes. So 2961 = 9 × 329. Now factorize 329. Let me check divisibility by 7: 7 × 47 = 329? 7 × 40 = 280, 7 × 7 = 49. 280 + 49 = 329. Yes. So 329 = 7 × 47. Therefore, 2961 = 9 × 7 × 47 = 3² × 7 × 47.

Now, factorizing 1491. Let me check divisibility by 3: 1 + 4 + 9 + 1 = 15, which is divisible by 3. 1491 ÷ 3 = 497. Let me check that: 3 × 497 = 1491. Now, factorize 497. Let me check divisibility by 7: 7 × 71 = 497? 7 × 70 = 490, plus 7 is 497. Yes! Therefore, 497 = 7 × 71. Therefore, 1491 = 3 × 7 × 71.

So, putting all together:

- 3339: 3² × 7 × 53
- 2961: 3² × 7 × 47
- 1491: 3 × 7 × 71

Now, the GCD is the product of the lowest power of common primes. Let's see. The common primes in all three numbers are 3 and 7.

For prime 3: in 3339 it's 3², in 2961 it's 3², and in 1491 it's 3¹. Therefore, the minimum exponent is 1.

For prime 7: all three numbers have 7¹. So, the minimum exponent is 1.

Therefore, GCD is 3¹ × 7¹ = 3 × 7 = 21.

So that confirms the earlier result. Therefore, 21 is indeed the greatest common divisor.

Wait, just to be thorough, let me check if there's a common divisor larger than 21. For example, could there be a common divisor of 21 × something? Let's see. Let me check if 21 × 2 = 42 divides all three numbers.

Check 3339 ÷ 42. 42 × 80 = 3360, which is higher than 3339. 42 × 79 = 42 × 80 - 42 = 3360 - 42 = 3318. 3339 - 3318 = 21. So remainder 21, so no. Therefore, 42 does not divide 3339.

Similarly, check 21 × 3 = 63. Let me check 3339 ÷ 63. Earlier, when we were computing GCD(3339, 2961), we found that GCD was 63, but after taking GCD with 1491, it became 21. Wait, but perhaps I need to check if 63 divides 1491? Let's see. 1491 ÷ 63. 63 × 23 = 1449. 1491 - 1449 = 42. So remainder 42, which means 63 does not divide 1491. Therefore, 63 is not a common divisor. Therefore, 21 is indeed the highest.

Alternatively, let me check if there is another common divisor. For example, let's see if there is a common divisor of 3, 7, or 21. Since we have already found that 21 is the GCD, and since 21 is the product of 3 and 7, and the prime factors are as above, there's no higher common factor.

So, seems like 21 is the correct answer.

Just to recap the steps:

1. Found GCD(3339, 2961) using Euclidean algorithm: 63.
2. Then found GCD(63, 1491) using Euclidean algorithm: 21.
3. Verified by prime factorization that the common factors are 3 and 7, so GCD is 3×7=21.
4. Checked that 21 divides all three numbers and that no higher number does.

Therefore, I'm confident that the greatest common divisor is 21.

**Final Answer**
The greatest common divisor of $3339$, $2961$, and $1491$ is \boxed{21}.

To find the **greatest common divisor (GCD)** of the numbers $ 3339 $, $ 2961 $, and $ 1491 $, we apply the **Euclidean algorithm** step by step. The Euclidean algorithm allows us to find the GCD of two numbers by repeatedly applying the remainder operation. If we want the GCD of three numbers, we first find the GCD of two of them and then find the GCD of that result with the third number.

---

### Step 1: Compute GCD(3339, 2961)

Use the Euclidean algorithm:

$$
\begin{align*}
3339 \div 2961 &= 1 \text{ with remainder } 378 \\
2961 \div 378 &= 7 \text{ with remainder } 315 \\
378 \div 315 &= 1 \text{ with remainder } 63 \\
315 \div 63 &= 5 \text{ with remainder } 0 \\
\end{align*}
$$

So, $ \gcd(3339, 2961) = 63 $

---

### Step 2: Compute GCD(63, 1491)

$$
\begin{align*}
1491 \div 63 &= 23 \text{ with remainder } 42 \\
63 \div 42 &= 1 \text{ with remainder } 21 \\
42 \div 21 &= 2 \text{ with remainder } 0 \\
\end{align*}
$$

So, $ \gcd(63, 1491) = 21 $

---

### Step 3: Interpret the Result

Since we now have:

$$
\gcd(3339, 2961, 1491) = \gcd(\gcd(3339, 2961), 1491) = \gcd(63, 1491) = 21
$$

We conclude that the **greatest common divisor** of the three numbers is:

$$
\boxed{21}
$$

Solution:
We can perform the Euclidean Algorithm two times.

First, we use it for $3339$ and $2961$. \begin{align*}
\text{gcd}\,(3339,2961) &=\text{gcd}\,(3339-2961,2961)\\
&=\text{gcd}\,(378,2961)\\
&=\text{gcd}\,(378,2961-378 \cdot 7)\\
&=\text{gcd}\,(378,315)\\
&=\text{gcd}\,(378-315,315)\\
&=\text{gcd}\,(63,315)\\
\end{align*}Since $63$ is a divisor of $315$, the greatest common divisor of $3339$ and $2961$ is $63$.

Next, we can find the greatest common divisor of $63$ and $1491$, also using the Euclidean Algorithm. \begin{align*}
\text{gcd}\,(63,1491) &=\text{gcd}\,(63,1491-63 \cdot 23)\\
&=\text{gcd}\,(63,42)\\
\end{align*}Since $63=3 \cdot 21$ and $42=2 \cdot 21$, the greatest common divisor is $\boxed{21}$.


Explanation:
Okay, so there's this pirate searching seven islands for buried treasure. Each island has a 1/5 chance of having treasure. The question is asking for the probability that exactly 4 of the islands have treasure. Hmm, let me think about how to approach this.

First, I remember that when dealing with probabilities of a certain number of successes in a fixed number of independent trials, it's a binomial probability problem. The formula for binomial probability is P(k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and C(n, k) is the combination of n things taken k at a time.

So in this case, the pirate is doing 7 trials (searching 7 islands), each with a success probability of 1/5 (finding treasure). We want exactly 4 successes. Let me plug these numbers into the formula.

First, I need to compute C(7, 4), which is the number of ways to choose 4 islands out of 7 where treasure is found. Then multiply that by (1/5)^4, which is the probability of finding treasure on those 4 islands, and then multiply by (4/5)^(7-4) = (4/5)^3, since the remaining 3 islands don't have treasure.

Wait, let me make sure. The probability of success is 1/5, so the probability of failure is 1 - 1/5 = 4/5. Right. So the formula is indeed C(7,4)*(1/5)^4*(4/5)^3.

Let me compute each part step by step.

First, C(7,4). The combination formula is C(n,k) = n! / (k! * (n - k)!). So plugging in 7 and 4:

C(7,4) = 7! / (4! * (7 - 4)!) = 7! / (4! * 3!)

Calculating 7! is 5040, 4! is 24, and 3! is 6. So 5040 / (24 * 6) = 5040 / 144. Let me divide 5040 by 144. Well, 144*35 = 5040, because 144*30=4320 and 144*5=720, so 4320+720=5040. Therefore, C(7,4) is 35.

Okay, so that part is done. Now, (1/5)^4. Let me compute that. 1/5 is 0.2, so 0.2^4. Let me compute that. 0.2 * 0.2 = 0.04, then * 0.2 = 0.008, then * 0.2 = 0.0016. Wait, so (1/5)^4 is 1/(5^4) = 1/625 = 0.0016. Correct.

Then, (4/5)^3. Let me calculate that. 4/5 is 0.8. 0.8^3. 0.8 * 0.8 = 0.64, then * 0.8 = 0.512. Alternatively, 4^3 / 5^3 = 64 / 125 = 0.512. Correct.

So putting it all together: 35 * (1/625) * (64/125). Wait, or maybe I should compute it as decimals. Let me see.

Alternatively, maybe it's easier to do fractions. Let me try:

First, 35 * (1/5)^4 * (4/5)^3 = 35 * (1^4 * 4^3) / (5^4 * 5^3) ) = 35 * (4^3) / (5^7). Let me compute that.

Wait, 5^7 is 5^4 * 5^3 = 625 * 125 = 78125. And 4^3 is 64. So 35 * 64 / 78125. Let me compute 35 * 64 first. 35 * 60 = 2100, 35 * 4 = 140, so total is 2100 + 140 = 2240. Therefore, 2240 / 78125. Let me check if that reduces. Let's see if 2240 and 78125 have a common factor. 2240 is 224*10, which is 2^5 * 5 * 7. Wait, 2240 = 2^6 * 5 * 7? Wait, 2240 divided by 2 is 1120, again 2 is 560, again 2 is 280, again 2 is 140, again 2 is 70, again 2 is 35. So that's 2^6 * 35. 35 is 5*7. So 2^6 * 5 *7.

And 78125 is 5^7, since 5^5 is 3125, 5^6 is 15625, 5^7 is 78125. So the numerator is 2^6 *5 *7 and the denominator is 5^7. So we can cancel one 5 from numerator and denominator. So that leaves 2^6 *7 / 5^6. Let me compute that.

Alternatively, maybe we can write it as a decimal. Let me divide 2240 by 78125. Let's see. 2240 divided by 78125. Let me write this as 2240 / 78125. Let me multiply numerator and denominator by 16 to make denominator a power of 10? Wait, maybe not. Alternatively, divide numerator and denominator by 5. 2240 divided by 5 is 448, and 78125 divided by 5 is 15625. So 448 / 15625. Again, divide numerator and denominator by... Let me check if 448 and 15625 have common factors. 448 is 2^6 *7, and 15625 is 5^6. So no common factors. Therefore, 448/15625 is the simplified fraction. If I convert that to a decimal, 448 divided by 15625. Let me do that division.

First, note that 15625 goes into 448 zero times. Let me write it as 448.0000 divided by 15625.

Alternatively, note that 1/15625 is 0.000064, because 1/10000 is 0.0001, so 1/15625 is less than that. Let me compute 448 * 0.000064. Wait, maybe another approach. Let me multiply numerator and denominator by 16 to make denominator 250000. Wait, 15625 * 16 = 250000. So 448 * 16 / 250000. Let me compute 448 * 16. 400*16=6400, 48*16=768. So total is 6400 + 768 = 7168. Therefore, 7168 / 250000. Simplify that. Divide numerator and denominator by 16: 7168 /16 = 448, 250000 /16 = 15625. Wait, we're back to the same fraction. Hmm.

Alternatively, let me do decimal division. Let me think of 448 divided by 15625. So since 15625 * 0.02 = 312.5, and 15625 * 0.028 = ?

Wait, maybe another way. Let me note that 1/15625 = 0.000064. Therefore, 448 * 0.000064. Let me compute that:

First, 400 * 0.000064 = 0.0256

Then 48 * 0.000064 = 0.003072

Adding together 0.0256 + 0.003072 = 0.028672

So the decimal is approximately 0.028672. So about 2.8672%.

Alternatively, if I use the original numbers:

35 * 0.0016 * 0.512. Let me calculate that.

First, 35 * 0.0016. Let's see: 35 * 0.0016. 0.0016 * 35 is same as 35 * 16 * 0.0001. 35*16 is 560, so 560 * 0.0001 = 0.056.

Then multiply that by 0.512: 0.056 * 0.512.

Let me compute 0.05 * 0.512 = 0.0256, and 0.006 * 0.512 = 0.003072. Adding them together: 0.0256 + 0.003072 = 0.028672. Same result as before. So decimal is approximately 0.028672, which is about 2.8672%.

So the exact probability is 2240/78125, which simplifies to 448/15625, which is approximately 0.028672.

But maybe the question expects the answer as a fraction or a decimal? The problem doesn't specify, but since it's probability, both are acceptable, but often fractions are preferred unless told otherwise. Let me check if 448/15625 can be simplified further. As mentioned before, 448 is 2^6 * 7 and 15625 is 5^6, so no common factors. Therefore, 448/15625 is simplest.

Alternatively, maybe we can write the original combination multiplied out:

C(7,4) is 35, so 35 * (1/5)^4 * (4/5)^3. Let me check if that is equal to 35 * 64 / 5^7. Wait, 5^7 is 78125, and 35*64 is 2240. Yes, same as before. So yeah, 2240/78125 reduces to 448/15625, so that is the exact probability.

Alternatively, if they want it as a decimal, we can write approximately 0.028672, which rounds to 0.0287 or 0.0287. But maybe the question wants an exact fraction. So 448/15625 is exact. Let me check if that fraction is correct.

Alternatively, let me confirm the calculation again:

C(7,4) = 35. Correct.

(1/5)^4 = 1/625. Correct.

(4/5)^3 = 64/125. Correct.

Multiply all together: 35 * (1/625) * (64/125) = 35*64/(625*125). 625 is 5^4, 125 is 5^3, so denominator is 5^7, which is 78125. Numerator is 35*64. 35*64: 35*60=2100, 35*4=140, total 2240. Therefore, yes, 2240/78125. Simplify by dividing numerator and denominator by 5: 448/15625. Correct.

Alternatively, maybe the problem expects the answer in terms of exponents or something else? But I think 448/15625 is the exact probability. Alternatively, if they want it as a decimal, maybe to four decimal places? Let me check the decimal value again.

As we calculated, 448 divided by 15625. Let me do this division more carefully.

So 15625 x 0.02 = 312.5

Subtract that from 448: 448 - 312.5 = 135.5

Bring down a zero (making it 1355.0). Now, how many times does 15625 go into 1355? 0 times. So 0.02 + 0.00...

Wait, maybe decimal division is tedious. Alternatively, use a calculator-like approach. Since I know that 1/15625 = 0.000064, then 448 * 0.000064 = ?

Let me compute 448 * 64 = ?

Well, 400*64 = 25,600

48*64 = 3,072

So total is 25,600 + 3,072 = 28,672. Then, since we have 0.000064, which is 64 * 10^{-6}, then multiplying 448 * 64 * 10^{-6} = 28,672 * 10^{-6} = 0.028672. Yes, so exactly 0.028672.

So, if we wanted to present the answer as a decimal, it would be approximately 0.028672, which is 2.8672%. But the question didn't specify, so maybe we should present the exact fraction, which is 448/15625. Alternatively, check if that fraction reduces further. Since 448 and 15625 have no common factors (as established earlier), that is the simplest form.

Alternatively, maybe the problem expects it in terms of the combination formula? But no, they just want the probability.

Wait, let me check once again if my initial approach was correct. The problem says "exactly 4 of the islands have treasure". So each island is independent, with probability 1/5 of having treasure, so yes, binomial distribution applies. So the formula is correct.

Another way to think about it: the number of ways to choose 4 islands out of 7 is C(7,4). For each of these choices, the probability that those 4 have treasure and the other 3 don't is (1/5)^4*(4/5)^3. So multiplying the number of ways by the probability for each case gives the total probability. Yep, that's exactly what I did. So the calculation seems correct.

So the answer is 448/15625, which is approximately 0.028672. Depending on what's required, but since the question says "put your final answer within \boxed{}", and math problems often prefer fractions unless specified otherwise, so I think \boxed{\dfrac{448}{15625}} is the answer. Alternatively, maybe they want it simplified differently or as a decimal? Let me check if 448/15625 can be simplified further. Since 448 is 2^6 * 7 and 15625 is 5^6, there are no common factors. Therefore, it is simplified.

Alternatively, if the problem expects a decimal, maybe rounding to a certain decimal place? But since it's exact, 448/15625 is better. Alternatively, maybe they want it written as a decimal with a fraction? Wait, no, the problem didn't specify. Let me check if 448/15625 is the correct exact value. Yes, as per calculation.

Alternatively, maybe I can check with another approach. For example, using the binomial probability formula, maybe with another calculation. Let me check the value of 35*(1/5)^4*(4/5)^3.

First, compute (1/5)^4 = 1/625 ≈ 0.0016

(4/5)^3 = 64/125 = 0.512

Multiply those two: 0.0016 * 0.512 = 0.0008192

Then multiply by 35: 0.0008192 * 35. Let me compute that.

First, 0.0008192 * 30 = 0.024576

Then, 0.0008192 * 5 = 0.004096

Adding them together: 0.024576 + 0.004096 = 0.028672. Yep, same result. So that's 0.028672, which is 448/15625 as a fraction.

Therefore, the exact probability is 448⁄15625, and the decimal is approximately 0.028672. Since the question doesn't specify the form, but given that it's a math competition-style problem, the fraction is likely expected.

So I think that's the answer. Let me just check once more if I made any computational errors in the combination or exponents.

C(7,4) is 35. Correct.

(1/5)^4 is 1/625, correct.

(4/5)^3 is 64/125, correct.

Multiplying all together: 35*(1/625)*(64/125). Let me compute denominator: 625*125 = 78125. Numerator: 35*64 = 2240. Yep. 2240/78125 reduces to 448/15625. Correct.

Yes, I think that's solid. So the probability is 448⁄15625.

**Final Answer**
The probability that exactly 4 of the islands have treasure is \boxed{\dfrac{448}{15625}}.

To find the probability that exactly 4 out of 7 islands contain treasure, we model this scenario using the **binomial distribution**.

---

### **Step 1: Understand the Scenario and Model the Problem**

Each of the 7 islands is an independent trial with:

- **Success (treasure found)** with probability $ p = \frac{1}{5} $
- **Failure (no treasure)** with probability $ 1 - p = \frac{4}{5} $

We are interested in the **probability of exactly 4 successes** (treasure discovery) in 7 trials.

This is a classic binomial probability problem, and we use the following formula:

$$
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
$$

Where:
- $ n = 7 $ (total number of trials)
- $ k = 4 $ (number of successes)
- $ p = \frac{1}{5} $
- $ 1 - p = \frac{4}{5} $

---

### **Step 2: Apply the Formula**

$$
P(X = 4) = \binom{7}{4} \left( \frac{1}{5} \right)^4 \left( \frac{4}{5} \right)^3
$$

**Step 2.1: Compute the combination**

$$
\binom{7}{4} = \frac{7!}{4! \cdot 3!} = \frac{5040}{24 \cdot 6} = \frac{5040}{144} = 35
$$

---

**Step 2.2: Compute the powers**

- $ \left( \frac{1}{5} \right)^4 = \frac{1}{625} $
- $ \left( \frac{4}{5} \right)^3 = \frac{64}{125} $

---

**Step 2.3: Multiply all parts together**

$$
P(X = 4) = 35 \cdot \frac{1}{625} \cdot \frac{64}{125}
$$

Convert to a single fraction:

$$
P(X = 4) = \frac{35 \cdot 64}{625 \cdot 125} = \frac{2240}{78125}
$$

Now reduce the fraction:

- $ \frac{2240}{78125} $ can be simplified by dividing numerator and denominator by 5:

$$
= \frac{448}{15625}
$$

This is the **simplest form** of the probability.

---

### **Final Answer**

$$
\boxed{\dfrac{448}{15625}}
$$

[Assistant]
Explanation:
Okay, so there's this pirate searching seven islands for buried treasure. Each island has a 1/5 chance of having treasure. The question is asking for the probability that exactly 4 of the islands have treasure. Hmm, let me think about how to approach this.

First, I remember that when dealing with probabilities of a certain number of successes in a fixed number of independent trials, it's a binomial probability problem. The formula for binomial probability is P(k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and C(n, k) is the combination of n things taken k at a time.

So in this case, the pirate is doing 7 trials (searching 7 islands), each with a success probability of 1/5 (finding treasure). We want exactly 4 successes. Let me plug these numbers into the formula.

First, I need to compute C(7, 4), which is the number of ways to choose 4 islands out of 7 where treasure is found. Then multiply that by (1/5)^4, which is the probability of finding treasure on those 4 islands, and then multiply by (4/5)^(7-4) = (4/5)^3, since the remaining 3 islands don't have treasure.

Wait, let me make sure. The probability of success is 1/5, so the probability of failure is 1 - 1/5 = 4/5. Right. So the formula is indeed C(7,4)*(1/5)^4*(4/5)^3.

Let me compute each part step by step.

First, C(7,4). The combination formula is C(n,k) = n! / (k! * (n - k)!). So plugging in 7 and 4:

C(7,4) = 7! / (4! * (7 - 4)!) = 7! / (4! * 3!)

Calculating 7! is 5040, 4! is 24, and 3! is 6. So 5040 / (24 * 6) = 5040 / 144. Let me divide 5040 by 144. Well, 144*35 = 5040, because 144*30=4320 and 144*5=720, so 4320+720=5040. Therefore, C(7,4) is 35.

Okay, so that part is done. Now, (1/5)^4. Let me compute that. 1/5 is 0.2, so 0.2^4. Let me compute that. 0.2 * 0.2 = 0.04, then * 0.2 = 0.008, then * 0.2 = 0.0016. Wait, so (1/5)^4 is 1/(5^4) = 1/625 = 0.0016. Correct.

Then, (4/5)^3. Let me calculate that. 4/5 is 0.8. 0.8^3. 0.8 * 0.8 = 0.64, then * 0.8 = 0.512. Alternatively, 4^3 / 5^3 = 64 / 125 = 0.512. Correct.

So putting it all together: 35 * (1/625) * (64/125). Wait, or maybe I should compute it as decimals. Let me see.

Alternatively, maybe it's easier to do fractions. Let me try:

First, 35 * (1/5)^4 * (4/5)^3 = 35 * (1^4 * 4^3) / (5^4 * 5^3) ) = 35 * (4^3) / (5^7). Let me compute that.

Wait, 5^7 is 5^4 * 5^3 = 625 * 125 = 78125. And 4^3 is 64. So 35 * 64 / 78125. Let me compute 35 * 64 first. 35 * 60 = 2100, 35 * 4 = 140, so total is 2100 + 140 = 2240. Therefore, 2240 / 78125. Let me check if that reduces. Let's see if 2240 and 78125 have a common factor. 2240 is 224*10, which is 2^5 * 5 * 7. Wait, 2240 = 2^6 * 5 * 7? Wait, 2240 divided by 2 is 1120, again 2 is 560, again 2 is 280, again 2 is 140, again 2 is 70, again 2 is 35. So that's 2^6 * 35. 35 is 5*7. So 2^6 * 5 *7.

And 78125 is 5^7, since 5^5 is 3125, 5^6 is 15625, 5^7 is 78125. So the numerator is 2^6 *5 *7 and the denominator is 5^7. So we can cancel one 5 from numerator and denominator. So that leaves 2^6 *7 / 5^6. Let me compute that.

Alternatively, maybe we can write it as a decimal. Let me divide 2240 by 78125. Let's see. 2240 divided by 78125. Let me write this as 2240 / 78125. Let me multiply numerator and denominator by 16 to make denominator a power of 10? Wait, maybe not. Alternatively, divide numerator and denominator by 5. 2240 divided by 5 is 448, and 78125 divided by 5 is 15625. So 448 / 15625. Again, divide numerator and denominator by... Let me check if 448 and 15625 have common factors. 448 is 2^6 *7, and 15625 is 5^6. So no common factors. Therefore, 448/15625 is the simplified fraction. If I convert that to a decimal, 448 divided by 15625. Let me do that division.

First, note that 15625 goes into 448 zero times. Let me write it as 448.0000 divided by 15625.

Alternatively, note that 1/15625 is 0.000064, because 1/10000 is 0.0001, so 1/15625 is less than that. Let me compute 448 * 0.000064. Wait, maybe another approach. Let me multiply numerator and denominator by 16 to make denominator 250000. Wait, 15625 * 16 = 250000. So 448 * 16 / 250000. Let me compute 448 * 16. 400*16=6400, 48*16=768. So total is 6400 + 768 = 7168. Therefore, 7168 / 250000. Simplify that. Divide numerator and denominator by 16: 7168 /16 = 448, 250000 /16 = 15625. Wait, we're back to the same fraction. Hmm.

Alternatively, let me do decimal division. Let me think of 448 divided by 15625. So since 15625 * 0.02 = 312.5, and 15625 * 0.028 = ?

Wait, maybe another way. Let me note that 1/15625 = 0.000064. Therefore, 448 * 0.000064. Let me compute that:

First, 400 * 0.000064 = 0.0256

Then 48 * 0.000064 = 0.003072

Adding together 0.0256 + 0.003072 = 0.028672

So the decimal is approximately 0.028672. So about 2.8672%.

Alternatively, if I use the original numbers:

35 * 0.0016 * 0.512. Let me calculate that.

First, 35 * 0.0016. Let's see: 35 * 0.0016. 0.0016 * 35 is same as 35 * 16 * 0.0001. 35*16 is 560, so 560 * 0.0001 = 0.056.

Then multiply that by 0.512: 0.056 * 0.512.

Let me compute 0.05 * 0.512 = 0.0256, and 0.006 * 0.512 = 0.003072. Adding them together: 0.0256 + 0.003072 = 0.028672. Same result as before. So decimal is approximately 0.028672, which is about 2.8672%.

So the exact probability is 2240/78125, which simplifies to 448/15625, which is approximately 0.028672.

But maybe the question expects the answer as a fraction or a decimal? The problem doesn't specify, but since it's probability, both are acceptable, but often fractions are preferred unless told otherwise. Let me check if 448/15625 can be simplified further. As mentioned before, 448 is 2^6 * 7 and 15625 is 5^6, so no common factors. Therefore, 448/15625 is simplest.

Alternatively, maybe we can write the original combination multiplied out:

C(7,4) is 35, so 35 * (1/5)^4 * (4/5)^3. Let me check if that is equal to 35 * 64 / 5^7. Wait, 5^7 is 78125, and 35*64 is 2240. Yes, same as before. So yeah, 2240/78125 reduces to 448/15625, so that is the exact probability.

Alternatively, if they want it as a decimal, we can write approximately 0.028672, which rounds to 0.0287 or 0.0287. But maybe the question wants an exact fraction. So 448/15625 is exact. Let me check if that fraction is correct.

Alternatively, let me confirm the calculation again:

C(7,4) = 35. Correct.

(1/5)^4 = 1/625. Correct.

(4/5)^3 = 64/125. Correct.

Multiply all together: 35 * (1/625) * (64/125) = 35*64/(625*125). 625 is 5^4, 125 is 5^3, so denominator is 5^7, which is 78125. Numerator is 35*64. 35*64: 35*60=2100, 35*4=140, total 2240. Therefore, yes, 2240/78125. Simplify by dividing numerator and denominator by 5: 448/15625. Correct.

Alternatively, maybe the problem expects the answer in terms of exponents or something else? But I think 448/15625 is the exact probability. Alternatively, if they want it as a decimal, maybe to four decimal places? Let me check the decimal value again.

As we calculated, 448 divided by 15625. Let me do this division more carefully.

So 15625 x 0.02 = 312.5

Subtract that from 448: 448 - 312.5 = 135.5

Bring down a zero (making it 1355.0). Now, how many times does 15625 go into 1355? 0 times. So 0.02 + 0.00...

Wait, maybe decimal division is tedious. Alternatively, use a calculator-like approach. Since I know that 1/15625 = 0.000064, then 448 * 0.000064 = ?

Let me compute 448 * 64 = ?

Well, 400*64 = 25,600

48*64 = 3,072

So total is 25,600 + 3,072 = 28,672. Then, since we have 0.000064, which is 64 * 10^{-6}, then multiplying 448 * 64 * 10^{-6} = 28,672 * 10^{-6} = 0.028672. Yes, so exactly 0.028672.

So, if we wanted to present the answer as a decimal, it would be approximately 0.028672, which is 2.8672%. But the question didn't specify, so maybe we should present the exact fraction, which is 448/15625. Alternatively, check if that fraction reduces further. Since 448 and 15625 have no common factors (as established earlier), that is the simplest form.

Alternatively, maybe the problem expects it in terms of the combination formula? But no, they just want the probability.

Wait, let me check once again if my initial approach was correct. The problem says "exactly 4 of the islands have treasure". So each island is independent, with probability 1/5 of having treasure, so yes, binomial distribution applies. So the formula is correct.

Another way to think about it: the number of ways to choose 4 islands out of 7 is C(7,4). For each of these choices, the probability that those 4 have treasure and the other 3 don't is (1/5)^4*(4/5)^3. So multiplying the number of ways by the probability for each case gives the total probability. Yep, that's exactly what I did. So the calculation seems correct.

So the answer is 448/15625, which is approximately 0.028672. Depending on what's required, but since the question says "put your final answer within \boxed{}", and math problems often prefer fractions unless specified otherwise, so I think \boxed{\dfrac{448}{15625}} is the answer. Alternatively, maybe they want it simplified differently or as a decimal? Let me check if 448/15625 can be simplified further. Since 448 is 2^6 * 7 and 15625 is 5^6, there are no common factors. Therefore, it is simplified.

Alternatively, if the problem expects a decimal, maybe rounding to a certain decimal place? But since it's exact, 448/15625 is better. Alternatively, maybe they want it written as a decimal with a fraction? Wait, no, the problem didn't specify. Let me check if 448/15625 is the correct exact value. Yes, as per calculation.

Alternatively, maybe I can check with another approach. For example, using the binomial probability formula, maybe with another calculation. Let me check the value of 35*(1/5)^4*(4/5)^3.

First, compute (1/5)^4 = 1/625 ≈ 0.0016

(4/5)^3 = 64/125 = 0.512

Multiply those two: 0.0016 * 0.512 = 0.0008192

Then multiply by 35: 0.0008192 * 35. Let me compute that.

First, 0.0008192 * 30 = 0.024576

Then, 0.0008192 * 5 = 0.004096

Adding them together: 0.024576 + 0.004096 = 0.028672. Yep, same result. So that's 0.028672, which is 448/15625 as a fraction.

Therefore, the exact probability is 448⁄15625, and the decimal is approximately 0.028672. Since the question doesn't specify the form, but given that it's a math competition-style problem, the fraction is likely expected.

So I think that's the answer. Let me just check once more if I made any computational errors in the combination or exponents.

C(7,4) is 35. Correct.

(1/5)^4 is 1/625, correct.

(4/5)^3 is 64/125, correct.

Multiplying all together: 35*(1/625)*(64/125). Let me compute denominator: 625*125 = 78125. Numerator: 35*64 = 2240. Yep. 2240/78125 reduces to 448/15625. Correct.

Yes, I think that's solid. So the probability is 448⁄15625.

**Final Answer**
The probability that exactly 4 of the islands have treasure is \boxed{\dfrac{448}{15625}}.

To find the probability that exactly 4 out of 7 islands contain treasure, we model this scenario using the **binomial distribution**.

---

### **Step 1: Understand the Scenario and Model the Problem**

Each of the 7 islands is an independent trial with:

- **Success (treasure found)** with probability $ p = \frac{1}{5} $
- **Failure (no treasure)** with probability $ 1 - p = \frac{4}{5} $

We are interested in the **probability of exactly 4 successes** (treasure discovery) in 7 trials.

This is a classic binomial probability problem, and we use the following formula:

$$
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
$$

Where:
- $ n = 7 $ (total number of trials)
- $ k = 4 $ (number of successes)
- $ p = \frac{1}{5} $
- $ 1 - p = \frac{4}{5} $

---

### **Step 2: Apply the Formula**

$$
P(X = 4) = \binom{7}{4} \left( \frac{1}{5} \right)^4 \left( \frac{4}{5} \right)^3
$$

**Step 2.1: Compute the combination**

$$
\binom{7}{4} = \frac{7!}{4! \cdot 3!} = \frac{5040}{24 \cdot 6} = \frac{5040}{144} = 35
$$

---

**Step 2.2: Compute the powers**

- $ \left( \frac{1}{5} \right)^4 = \frac{1}{625} $
- $ \left( \frac{4}{5} \right)^3 = \frac{64}{125} $

---

**Step 2.3: Multiply all parts together**

$$
P(X = 4) = 35 \cdot \frac{1}{625} \cdot \frac{64}{125}
$$

Convert to a single fraction:

$$
P(X = 4) = \frac{35 \cdot 64}{625 \cdot 125} = \frac{2240}{78125}
$$

Now reduce the fraction:

- $ \frac{2240}{78125} $ can be simplified by dividing numerator and denominator by 5:

$$
= \frac{448}{15625}
$$

This is the **simplest form** of the probability.

---

### **Final Answer**

$$
\boxed{\dfrac{448}{15625}}
$$

Solution: There are $\binom{7}{4}=35$ ways to choose 4 of the islands. For each choice, there is a probability of $\left( \frac{1}{5} \right)^4 \left( \frac{4}{5} \right)^3$ that those 4 islands will have treasure and the others will not. Therefore, the total probability that exactly 4 of the islands have treasure is $35 \left( \frac{1}{5} \right)^4 \left( \frac{4}{5} \right)^3 = \boxed{\frac{448}{15625}}$.


Explanation:
Okay, so I need to find the value of c that makes the radius of the circle given by the equation x² - 10x + y² + 6y + c = 0 equal to 1. Hmm, let me recall how to handle the general equation of a circle.

First, the standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. The given equation isn't in standard form, so I think I need to complete the square for both the x and y terms.

Starting with the given equation: x² - 10x + y² + 6y + c = 0. Let me rearrange the terms to group the x terms and y terms together:

(x² - 10x) + (y² + 6y) + c = 0.

Now, I need to complete the square for each variable.

Starting with the x terms: x² - 10x. To complete the square, I take the coefficient of x, which is -10, divide by 2 to get -5, and then square it to get 25. So, I add and subtract 25 to the equation.

Similarly, for the y terms: y² + 6y. The coefficient of y is 6, divide by 2 to get 3, square it to get 9. So I add and subtract 9 here as well.

But wait, I need to be careful here because the original equation has a constant term c. When I add these numbers (25 and 9) to complete the square, I have to balance the equation. Let me write this step by step.

Starting with the original equation:

x² - 10x + y² + 6y + c = 0.

Group the x and y terms:

(x² - 10x) + (y² + 6y) = -c.

Now, complete the square for x:

Take x² - 10x, add 25 to complete the square:

(x² - 10x + 25) - 25.

Similarly for y² + 6y, add 9:

(y² + 6y + 9) - 9.

So substituting back into the equation:

(x² - 10x + 25) - 25 + (y² + 6y + 9) - 9 = -c.

Simplify:

(x - 5)² + (y + 3)² - 25 - 9 = -c.

Which becomes:

(x - 5)² + (y + 3)² - 34 = -c.

Then, moving the -34 to the right side:

(x - 5)² + (y + 3)² = -c + 34.

So the standard form of the circle is (x - 5)² + (y + 3)² = 34 - c.

Now, the radius squared is equal to 34 - c. Since the radius is given as 1, then the radius squared is 1² = 1.

Therefore, setting up the equation:

34 - c = 1.

Solving for c:

c = 34 - 1 = 33.

Wait, that seems straightforward. Let me check again.

Original equation:

x² -10x + y² +6y +c =0.

Complete the squares:

x² -10x becomes (x-5)^2 -25

y² +6y becomes (y+3)^2 -9

So substituting back:

(x-5)^2 -25 + (y+3)^2 -9 + c = 0.

Combine constants:

(x-5)^2 + (y+3)^2 -34 + c = 0.

Therefore,

(x-5)^2 + (y+3)^2 = 34 - c.

Yes, same as before. Then radius squared is 34 - c. If radius is 1, then 34 - c = 1, so c = 34 - 1 = 33.

So c = 33?

But wait, let me verify by plugging back into the original equation. Let me see.

Suppose c = 33. Then original equation becomes:

x² -10x + y² +6y +33 = 0.

Completing the squares:

(x² -10x +25) + (y² +6y +9) = -33 +25 +9.

Which is (x -5)^2 + (y +3)^2 = 1.

Yes! That is a circle with center (5, -3) and radius 1. So that checks out. Therefore, c = 33 is correct.

But just to make sure I didn't make any sign errors. Let me check the completion of squares again.

Original x terms: x² -10x. Completing the square: take half of -10, which is -5, square it is 25. So x² -10x +25 is (x-5)^2. Therefore, we have to add 25. But since in the original equation the x terms are x² -10x, when we add 25, we need to subtract it as well to keep the equation balanced. Wait, no. Wait, when we complete the square, we have:

Original equation after grouping:

(x² -10x) + (y² +6y) + c = 0.

Then, we add 25 and 9 to complete the squares, so we have:

(x² -10x +25) + (y² +6y +9) + c = 0 +25 +9.

Wait, is that correct? Wait, no. Because when you add 25 and 9 to complete the square, you have to do it on both sides of the equation. Let me think again.

Original equation:

(x² -10x) + (y² +6y) + c = 0.

To complete the square for x, add 25 to both sides:

(x² -10x +25) + (y² +6y) + c = 25.

Similarly, complete the square for y by adding 9 to both sides:

(x² -10x +25) + (y² +6y +9) + c = 25 +9.

Then, that gives:

(x -5)^2 + (y +3)^2 + c = 34.

Then, moving c to the other side:

(x -5)^2 + (y +3)^2 = 34 - c.

Yes, same as before. So if radius squared is 1, then 34 - c = 1, so c = 33.

So that seems correct. Therefore, the answer is 33.

But just to be thorough, maybe I can check with another method. Let me recall that the general equation of a circle is x² + y² + Dx + Ey + F = 0, which can be rewritten as (x + D/2)^2 + (y + E/2)^2 = (D/2)^2 + (E/2)^2 - F.

So in this case, the given equation is x² -10x + y² +6y +c = 0. Comparing to the general equation, D is -10, E is 6, F is c.

Then the radius squared is (D/2)^2 + (E/2)^2 - F.

So that would be (-10/2)^2 + (6/2)^2 - c = (-5)^2 + 3^2 - c = 25 + 9 - c = 34 - c.

Which is the same as before. Therefore, radius squared is 34 - c, set equal to 1. Therefore, 34 - c = 1 => c = 33. Yep, same answer.

So that seems consistent. Therefore, I think c is 33.

But let me think again if there is any possible mistake here. For example, sometimes when completing the square, people make sign errors. Let me check the signs again.

Original equation:

x² -10x + y² +6y + c = 0.

To complete the square for x² -10x: take half of -10, which is -5, square it is 25. Therefore, x² -10x = (x - 5)^2 - 25. Similarly, y² +6y: half of 6 is 3, square is 9, so y² +6y = (y +3)^2 -9.

Therefore, substituting back into the original equation:

(x -5)^2 -25 + (y +3)^2 -9 + c = 0.

So combining constants: -25 -9 = -34, so:

(x -5)^2 + (y +3)^2 -34 + c = 0 => (x -5)^2 + (y +3)^2 = 34 - c.

Yes, correct. So radius squared is 34 - c = 1, so c = 34 -1 =33.

Therefore, all checks out.

Alternatively, if I think of the radius formula for general circle equation. The radius is sqrt( (D/2)^2 + (E/2)^2 - F ). So in this case, radius is sqrt( (-10/2)^2 + (6/2)^2 - c ) = sqrt(25 + 9 - c) = sqrt(34 -c).

Set that equal to 1: sqrt(34 -c) = 1. Square both sides: 34 -c = 1 => c = 33.

Same result. Therefore, I think this is correct.

I think I have verified this multiple times through different methods: completing the square, using the general formula for radius, and plugging back in. All lead to the same answer, c = 33. So I think that is correct.

Just to ensure that there isn't a negative sign somewhere that I missed. For example, when moving terms around, but no. The equation after completing the square was (x -5)^2 + (y +3)^2 = 34 - c. So if c is 33, then 34 -33 = 1, which is the radius squared. So that's correct.

Alternatively, if someone thought that the radius squared is c - 34, then they would get a different answer, but that would be wrong. But since the equation is (x - h)^2 + (y - k)^2 = r^2, then the right-hand side has to be positive. So 34 - c must be positive. Since radius squared is 1, which is positive, then 34 - c must be positive, so c < 34. And since c is 33, which is less than 34, that is okay.

Therefore, everything seems to check out.

**Final Answer**
The value of $ c $ is \boxed{33}.

To determine the value of $ c $ that makes the radius of the given circle equal to 1, we begin by analyzing the general form of the circle equation:

$$
x^2 - 10x + y^2 + 6y + c = 0
$$

---

### Step 1: Complete the Square

We rewrite this equation by completing the square for both $ x $ and $ y $.

#### For the $ x $-terms:
$$
x^2 - 10x = (x - 5)^2 - 25
$$

#### For the $ y $-terms:
$$
y^2 + 6y = (y + 3)^2 - 9
$$

Substitute these back into the original equation:

$$
(x - 5)^2 - 25 + (y + 3)^2 - 9 + c = 0
$$

Simplify the constants:

$$
(x - 5)^2 + (y + 3)^2 - 34 + c = 0
$$

Move the constants to the right side:

$$
(x - 5)^2 + (y + 3)^2 = 34 - c
$$

---

### Step 2: Use the Standard Form of a Circle

The standard form of a circle is:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

Comparing to the equation we obtained:

- Center: $ (5, -3) $
- Radius squared: $ r^2 = 34 - c $

We are told that the radius is 1. Therefore, we set:

$$
r^2 = 1 \Rightarrow 34 - c = 1
$$

---

### Step 3: Solve for $ c $

$$
34 - c = 1 \Rightarrow c = 34 - 1 = 33
$$

---

### Final Answer

$$
\boxed{33}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of c that makes the radius of the circle given by the equation x² - 10x + y² + 6y + c = 0 equal to 1. Hmm, let me recall how to handle the general equation of a circle.

First, the standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. The given equation isn't in standard form, so I think I need to complete the square for both the x and y terms.

Starting with the given equation: x² - 10x + y² + 6y + c = 0. Let me rearrange the terms to group the x terms and y terms together:

(x² - 10x) + (y² + 6y) + c = 0.

Now, I need to complete the square for each variable.

Starting with the x terms: x² - 10x. To complete the square, I take the coefficient of x, which is -10, divide by 2 to get -5, and then square it to get 25. So, I add and subtract 25 to the equation.

Similarly, for the y terms: y² + 6y. The coefficient of y is 6, divide by 2 to get 3, square it to get 9. So I add and subtract 9 here as well.

But wait, I need to be careful here because the original equation has a constant term c. When I add these numbers (25 and 9) to complete the square, I have to balance the equation. Let me write this step by step.

Starting with the original equation:

x² - 10x + y² + 6y + c = 0.

Group the x and y terms:

(x² - 10x) + (y² + 6y) = -c.

Now, complete the square for x:

Take x² - 10x, add 25 to complete the square:

(x² - 10x + 25) - 25.

Similarly for y² + 6y, add 9:

(y² + 6y + 9) - 9.

So substituting back into the equation:

(x² - 10x + 25) - 25 + (y² + 6y + 9) - 9 = -c.

Simplify:

(x - 5)² + (y + 3)² - 25 - 9 = -c.

Which becomes:

(x - 5)² + (y + 3)² - 34 = -c.

Then, moving the -34 to the right side:

(x - 5)² + (y + 3)² = -c + 34.

So the standard form of the circle is (x - 5)² + (y + 3)² = 34 - c.

Now, the radius squared is equal to 34 - c. Since the radius is given as 1, then the radius squared is 1² = 1.

Therefore, setting up the equation:

34 - c = 1.

Solving for c:

c = 34 - 1 = 33.

Wait, that seems straightforward. Let me check again.

Original equation:

x² -10x + y² +6y +c =0.

Complete the squares:

x² -10x becomes (x-5)^2 -25

y² +6y becomes (y+3)^2 -9

So substituting back:

(x-5)^2 -25 + (y+3)^2 -9 + c = 0.

Combine constants:

(x-5)^2 + (y+3)^2 -34 + c = 0.

Therefore,

(x-5)^2 + (y+3)^2 = 34 - c.

Yes, same as before. Then radius squared is 34 - c. If radius is 1, then 34 - c = 1, so c = 34 - 1 = 33.

So c = 33?

But wait, let me verify by plugging back into the original equation. Let me see.

Suppose c = 33. Then original equation becomes:

x² -10x + y² +6y +33 = 0.

Completing the squares:

(x² -10x +25) + (y² +6y +9) = -33 +25 +9.

Which is (x -5)^2 + (y +3)^2 = 1.

Yes! That is a circle with center (5, -3) and radius 1. So that checks out. Therefore, c = 33 is correct.

But just to make sure I didn't make any sign errors. Let me check the completion of squares again.

Original x terms: x² -10x. Completing the square: take half of -10, which is -5, square it is 25. So x² -10x +25 is (x-5)^2. Therefore, we have to add 25. But since in the original equation the x terms are x² -10x, when we add 25, we need to subtract it as well to keep the equation balanced. Wait, no. Wait, when we complete the square, we have:

Original equation after grouping:

(x² -10x) + (y² +6y) + c = 0.

Then, we add 25 and 9 to complete the squares, so we have:

(x² -10x +25) + (y² +6y +9) + c = 0 +25 +9.

Wait, is that correct? Wait, no. Because when you add 25 and 9 to complete the square, you have to do it on both sides of the equation. Let me think again.

Original equation:

(x² -10x) + (y² +6y) + c = 0.

To complete the square for x, add 25 to both sides:

(x² -10x +25) + (y² +6y) + c = 25.

Similarly, complete the square for y by adding 9 to both sides:

(x² -10x +25) + (y² +6y +9) + c = 25 +9.

Then, that gives:

(x -5)^2 + (y +3)^2 + c = 34.

Then, moving c to the other side:

(x -5)^2 + (y +3)^2 = 34 - c.

Yes, same as before. So if radius squared is 1, then 34 - c = 1, so c = 33.

So that seems correct. Therefore, the answer is 33.

But just to be thorough, maybe I can check with another method. Let me recall that the general equation of a circle is x² + y² + Dx + Ey + F = 0, which can be rewritten as (x + D/2)^2 + (y + E/2)^2 = (D/2)^2 + (E/2)^2 - F.

So in this case, the given equation is x² -10x + y² +6y +c = 0. Comparing to the general equation, D is -10, E is 6, F is c.

Then the radius squared is (D/2)^2 + (E/2)^2 - F.

So that would be (-10/2)^2 + (6/2)^2 - c = (-5)^2 + 3^2 - c = 25 + 9 - c = 34 - c.

Which is the same as before. Therefore, radius squared is 34 - c, set equal to 1. Therefore, 34 - c = 1 => c = 33. Yep, same answer.

So that seems consistent. Therefore, I think c is 33.

But let me think again if there is any possible mistake here. For example, sometimes when completing the square, people make sign errors. Let me check the signs again.

Original equation:

x² -10x + y² +6y + c = 0.

To complete the square for x² -10x: take half of -10, which is -5, square it is 25. Therefore, x² -10x = (x - 5)^2 - 25. Similarly, y² +6y: half of 6 is 3, square is 9, so y² +6y = (y +3)^2 -9.

Therefore, substituting back into the original equation:

(x -5)^2 -25 + (y +3)^2 -9 + c = 0.

So combining constants: -25 -9 = -34, so:

(x -5)^2 + (y +3)^2 -34 + c = 0 => (x -5)^2 + (y +3)^2 = 34 - c.

Yes, correct. So radius squared is 34 - c = 1, so c = 34 -1 =33.

Therefore, all checks out.

Alternatively, if I think of the radius formula for general circle equation. The radius is sqrt( (D/2)^2 + (E/2)^2 - F ). So in this case, radius is sqrt( (-10/2)^2 + (6/2)^2 - c ) = sqrt(25 + 9 - c) = sqrt(34 -c).

Set that equal to 1: sqrt(34 -c) = 1. Square both sides: 34 -c = 1 => c = 33.

Same result. Therefore, I think this is correct.

I think I have verified this multiple times through different methods: completing the square, using the general formula for radius, and plugging back in. All lead to the same answer, c = 33. So I think that is correct.

Just to ensure that there isn't a negative sign somewhere that I missed. For example, when moving terms around, but no. The equation after completing the square was (x -5)^2 + (y +3)^2 = 34 - c. So if c is 33, then 34 -33 = 1, which is the radius squared. So that's correct.

Alternatively, if someone thought that the radius squared is c - 34, then they would get a different answer, but that would be wrong. But since the equation is (x - h)^2 + (y - k)^2 = r^2, then the right-hand side has to be positive. So 34 - c must be positive. Since radius squared is 1, which is positive, then 34 - c must be positive, so c < 34. And since c is 33, which is less than 34, that is okay.

Therefore, everything seems to check out.

**Final Answer**
The value of $ c $ is \boxed{33}.

To determine the value of $ c $ that makes the radius of the given circle equal to 1, we begin by analyzing the general form of the circle equation:

$$
x^2 - 10x + y^2 + 6y + c = 0
$$

---

### Step 1: Complete the Square

We rewrite this equation by completing the square for both $ x $ and $ y $.

#### For the $ x $-terms:
$$
x^2 - 10x = (x - 5)^2 - 25
$$

#### For the $ y $-terms:
$$
y^2 + 6y = (y + 3)^2 - 9
$$

Substitute these back into the original equation:

$$
(x - 5)^2 - 25 + (y + 3)^2 - 9 + c = 0
$$

Simplify the constants:

$$
(x - 5)^2 + (y + 3)^2 - 34 + c = 0
$$

Move the constants to the right side:

$$
(x - 5)^2 + (y + 3)^2 = 34 - c
$$

---

### Step 2: Use the Standard Form of a Circle

The standard form of a circle is:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

Comparing to the equation we obtained:

- Center: $ (5, -3) $
- Radius squared: $ r^2 = 34 - c $

We are told that the radius is 1. Therefore, we set:

$$
r^2 = 1 \Rightarrow 34 - c = 1
$$

---

### Step 3: Solve for $ c $

$$
34 - c = 1 \Rightarrow c = 34 - 1 = 33
$$

---

### Final Answer

$$
\boxed{33}
$$

Solution: Completing the square gives us $(x - 5)^2 + (y + 3)^2 = 34 - c$. Since we want the radius to be 1, we must have $34 - c = 1^2$. It follows that $c = \boxed{33}$.