Explanation:
Okay, so I need to find the values of 'a' and 'b' such that when I multiply the two quadratic expressions (3t² + 5t + a) and (4t² + bt - 2), the result is the given quartic polynomial 12t⁴ + 26t³ - 8t² - 16t + 6. Then, once I find 'a' and 'b', I have to add them together to get the answer. Alright, let me start by recalling how to multiply two polynomials.

First, I know that when you multiply two polynomials, you have to use the distributive property, which means each term in the first polynomial gets multiplied by each term in the second polynomial. So, let me write down the two polynomials:

First polynomial: 3t² + 5t + a

Second polynomial: 4t² + bt - 2

So, multiplying them together would give me:

(3t²)(4t²) + (3t²)(bt) + (3t²)(-2) + (5t)(4t²) + (5t)(bt) + (5t)(-2) + (a)(4t²) + (a)(bt) + (a)(-2)

Let me compute each term step by step.

Starting with the first term: 3t² multiplied by 4t². Multiplying the coefficients: 3*4 = 12. Then, multiplying the variables: t²*t² = t⁴. So, that term is 12t⁴.

Next term: 3t² multiplied by bt. Coefficients: 3*b = 3b. Variables: t²*t = t³. So, that term is 3b t³.

Third term: 3t² multiplied by -2. Coefficients: 3*(-2) = -6. Variables: t². So, that term is -6t².

Fourth term: 5t multiplied by 4t². Coefficients: 5*4 = 20. Variables: t*t² = t³. So, that term is 20t³.

Fifth term: 5t multiplied by bt. Coefficients: 5*b = 5b. Variables: t*t = t². So, that term is 5b t².

Sixth term: 5t multiplied by -2. Coefficients: 5*(-2) = -10. Variables: t. So, that term is -10t.

Seventh term: a multiplied by 4t². Coefficients: a*4 = 4a. Variables: t². So, that term is 4a t².

Eighth term: a multiplied by bt. Coefficients: a*b = ab. Variables: t. So, that term is ab t.

Ninth term: a multiplied by -2. Coefficients: a*(-2) = -2a. Variables: none, so just a constant term. So, that term is -2a.

Now, let me write all these terms together:

12t⁴ + 3b t³ - 6t² + 20t³ + 5b t² - 10t + 4a t² + ab t - 2a.

Now, I need to combine like terms. Let me group them by their degree:

- For t⁴: only 12t⁴.

- For t³: 3b t³ + 20t³. Let me factor out t³: (3b + 20) t³.

- For t²: -6t² + 5b t² + 4a t². Let me factor out t²: (-6 + 5b + 4a) t².

- For t: -10t + ab t. Factor out t: (-10 + ab) t.

- Constants: -2a.

So, putting it all together, the product is:

12t⁴ + (3b + 20) t³ + (-6 + 5b + 4a) t² + (-10 + ab) t - 2a.

Now, according to the problem, this product is equal to 12t⁴ + 26t³ - 8t² - 16t + 6. Therefore, the coefficients of corresponding terms must be equal. Let me set up equations for each coefficient.

Starting with the t⁴ term:

Coefficient of t⁴: 12 = 12. Well, that's already matching, so no new information here.

Next, the coefficient of t³: 3b + 20 = 26.

Then, the coefficient of t²: -6 + 5b + 4a = -8.

Coefficient of t: -10 + ab = -16.

Constant term: -2a = 6.

So, now I have four equations:

1) 3b + 20 = 26

2) -6 + 5b + 4a = -8

3) -10 + ab = -16

4) -2a = 6

Let me solve these equations step by step.

Starting with equation 4: -2a = 6. Solving for a:

Divide both sides by -2: a = 6 / (-2) = -3. So, a = -3.

Okay, so a is -3. Let me note that down: a = -3.

Now, let me plug this value of a into equation 2 and equation 3.

First, equation 2: -6 + 5b + 4a = -8.

Substituting a = -3:

-6 + 5b + 4*(-3) = -8

Compute 4*(-3) = -12.

So, equation becomes: -6 + 5b -12 = -8.

Combine constants: -6 -12 = -18.

So, -18 + 5b = -8.

Now, solving for b:

Add 18 to both sides: 5b = -8 + 18 = 10.

Therefore, 5b = 10 => b = 10 / 5 = 2. So, b = 2.

Alright, so b is 2. Let me check equation 1 to make sure.

Equation 1: 3b + 20 = 26.

Substituting b = 2: 3*2 + 20 = 6 + 20 = 26. Which matches. Good.

Now, let me check equation 3: -10 + ab = -16.

We have a = -3 and b = 2. So, ab = (-3)(2) = -6.

Therefore, -10 + (-6) = -16. Which is -16 = -16. Perfect, that checks out.

So, all equations are satisfied with a = -3 and b = 2. Therefore, the question asks for a + b. So, a + b = (-3) + 2 = -1.

Wait, but let me just make sure I didn't make any calculation errors. Let me verify the entire multiplication with these values of a and b to ensure that the product is indeed the given polynomial.

So, substituting a = -3 and b = 2 into the original polynomials:

First polynomial: 3t² + 5t + a = 3t² + 5t - 3.

Second polynomial: 4t² + bt - 2 = 4t² + 2t - 2.

Now, let me multiply these two polynomials step by step.

First, multiply 3t² by each term in the second polynomial:

3t² * 4t² = 12t⁴

3t² * 2t = 6t³

3t² * (-2) = -6t²

Next, multiply 5t by each term in the second polynomial:

5t * 4t² = 20t³

5t * 2t = 10t²

5t * (-2) = -10t

Then, multiply -3 by each term in the second polynomial:

-3 * 4t² = -12t²

-3 * 2t = -6t

-3 * (-2) = 6

Now, let me combine all these terms:

12t⁴ + 6t³ -6t² + 20t³ + 10t² -10t -12t² -6t + 6.

Now, combine like terms:

t⁴ term: 12t⁴

t³ terms: 6t³ + 20t³ = 26t³

t² terms: -6t² + 10t² -12t² = (-6 + 10 -12)t² = (-8)t²

t terms: -10t -6t = -16t

Constants: +6

So, altogether, the product is 12t⁴ + 26t³ -8t² -16t +6, which matches exactly the given polynomial. Perfect! So, my values for a and b are correct.

Therefore, a + b = -3 + 2 = -1.

But wait, the problem says "What is a + b?" So, the answer is -1? Let me check once again if I didn't mix up any signs.

Original equations:

From equation 4: -2a = 6 => a = -3. Correct.

Equation 1: 3b + 20 = 26 => 3b = 6 => b = 2. Correct.

Equation 2: -6 + 5b + 4a = -8. Plugging in a=-3, b=2:

-6 + 5*2 + 4*(-3) = -6 + 10 -12 = (-6 -12) +10 = -18 +10 = -8. Correct.

Equation 3: -10 + ab = -16. ab = (-3)(2) = -6. -10 + (-6) = -16. Correct.

All equations are satisfied. So, a + b = -3 + 2 = -1. Therefore, the answer is -1.

But let me check once again the multiplication step-by-step to be absolutely sure. Maybe I made a mistake in combining terms?

First polynomial: 3t² +5t -3

Second polynomial: 4t² +2t -2

Multiplying term by term:

First, 3t²*(4t² +2t -2) = 12t⁴ +6t³ -6t²

Then, 5t*(4t² +2t -2) = 20t³ +10t² -10t

Then, -3*(4t² +2t -2) = -12t² -6t +6

Adding all together:

12t⁴ +6t³ -6t² +20t³ +10t² -10t -12t² -6t +6

Combine t⁴: 12t⁴

t³: 6t³ +20t³ = 26t³

t²: -6t² +10t² -12t² = (-6 +10 -12)t² = (-8)t²

t terms: -10t -6t = -16t

Constants: +6

Yes, exactly matches. So, no mistake here. Therefore, the answer is indeed -1.

But just to make sure, let me check if there's another way to approach this problem, maybe by comparing coefficients differently or using another method. For example, maybe plugging in specific values of t to create equations? Although since we already have a consistent solution, maybe that's redundant, but just for thoroughness.

Alternatively, suppose I set t = 0. Let me try that. If t = 0, then the original product is:

(3*(0)^2 +5*(0) +a)*(4*(0)^2 +b*(0) -2) = (0 + 0 +a)*(0 + 0 -2) = a*(-2) = -2a.

On the other hand, the given polynomial evaluated at t=0 is 12*(0)^4 +26*(0)^3 -8*(0)^2 -16*(0) +6 = 0 + 0 -0 -0 +6 = 6.

Therefore, -2a = 6 => a = -3. Which matches our previous result. So that's a good check.

Similarly, maybe plug in t = 1? Let me try that.

Original product at t=1:

(3*(1)^2 +5*(1) +a)*(4*(1)^2 +b*(1) -2) = (3 +5 +a)*(4 +b -2) = (8 +a)*(2 +b)

Given polynomial at t=1: 12*(1)^4 +26*(1)^3 -8*(1)^2 -16*(1) +6 = 12 +26 -8 -16 +6 = Let's compute:

12 +26 = 38; 38 -8 = 30; 30 -16 = 14; 14 +6 = 20.

Therefore, (8 +a)(2 +b) = 20.

We already know that a = -3, so substituting:

(8 + (-3))*(2 + b) = (5)*(2 + b) = 20 => 5*(2 + b) = 20 => 2 + b = 4 => b = 2. Which again matches our previous result. So that's another check.

Alternatively, maybe plug in t = -1? Let me try that for thoroughness.

Original product at t=-1:

(3*(-1)^2 +5*(-1) +a)*(4*(-1)^2 +b*(-1) -2) = (3*1 -5 +a)*(4*1 -b -2) = (3 -5 +a)*(4 -b -2) = (-2 +a)*(2 -b)

Given polynomial at t=-1:

12*(-1)^4 +26*(-1)^3 -8*(-1)^2 -16*(-1) +6 = 12*1 +26*(-1) -8*1 -16*(-1) +6 = 12 -26 -8 +16 +6.

Calculating step by step:

12 -26 = -14; -14 -8 = -22; -22 +16 = -6; -6 +6 = 0.

Therefore, (-2 +a)*(2 -b) = 0.

We know a = -3 and b = 2, so substituting:

(-2 + (-3))*(2 - 2) = (-5)*(0) = 0. Which matches. So that's correct.

So, all these checks confirm that a = -3 and b = 2, so a + b = -1. Therefore, I can be confident that the answer is -1.

But just to make sure, let me check if there's any other possible solutions. For example, suppose there was a case where equations could have multiple solutions, but in this case, since we had four equations and two variables, but actually, the equations were not all independent. Let me check the number of equations.

We had four equations, but actually, the first equation (from t⁴ term) was redundant because it was already satisfied. Then, the other three equations (from t³, t², t, and constant term) gave us three equations. However, since we had two variables, a and b, we needed two equations. But in reality, we had four equations, but they were not all independent. Let me check:

Equation 1: 3b + 20 = 26 => 3b = 6 => b = 2.

Equation 4: -2a = 6 => a = -3.

Then equations 2 and 3 were dependent on these values. So, in reality, equations 1 and 4 gave us the values of a and b directly, and equations 2 and 3 were just checks. Therefore, there is a unique solution here. Therefore, there are no other solutions. So, the answer is definitely -1.

Therefore, I think I can conclude that a + b = -1.

**Final Answer**
The value of $a + b$ is \boxed{-1}.

To solve the problem, we are given two quadratic polynomials:

$$
(3t^2 + 5t + a)(4t^2 + bt - 2)
$$

and their product is:

$$
12t^4 + 26t^3 - 8t^2 - 16t + 6
$$

Our goal is to find the values of $ a $ and $ b $, and then compute $ a + b $.

---

### Step 1: Multiply the Polynomials

We expand the product:

$$
(3t^2 + 5t + a)(4t^2 + bt - 2)
$$

Using the distributive property, we compute each term:

- $ 3t^2 \cdot 4t^2 = 12t^4 $
- $ 3t^2 \cdot bt = 3bt^3 $
- $ 3t^2 \cdot (-2) = -6t^2 $
- $ 5t \cdot 4t^2 = 20t^3 $
- $ 5t \cdot bt = 5bt^2 $
- $ 5t \cdot (-2) = -10t $
- $ a \cdot 4t^2 = 4at^2 $
- $ a \cdot bt = abt $
- $ a \cdot (-2) = -2a $

Now, combine like terms:

- $ t^4 $: $ 12t^4 $
- $ t^3 $: $ (3b + 20)t^3 $
- $ t^2 $: $ (-6 + 5b + 4a)t^2 $
- $ t $: $ (-10 + ab)t $
- Constant: $ -2a $

So the product becomes:

$$
12t^4 + (3b + 20)t^3 + (-6 + 5b + 4a)t^2 + (-10 + ab)t - 2a
$$

---

### Step 2: Match Coefficients

We are told this equals:

$$
12t^4 + 26t^3 - 8t^2 - 16t + 6
$$

Now, equate the coefficients of corresponding powers of $ t $:

1. $ 12 = 12 $ (already matches)
2. $ 3b + 20 = 26 $ → $ 3b = 6 $ → $ b = 2 $
3. $ -6 + 5b + 4a = -8 $
4. $ -10 + ab = -16 $
5. $ -2a = 6 $ → $ a = -3 $

---

### Step 3: Solve for $ a $ and $ b $

From equation (5):
$$
-2a = 6 \Rightarrow a = -3
$$

Substitute $ a = -3 $ into equation (3):
$$
-6 + 5b + 4(-3) = -8 \Rightarrow -6 + 5b - 12 = -8 \Rightarrow 5b = 10 \Rightarrow b = 2
$$

Now verify with equation (4):
$$
-10 + ab = -10 + (-3)(2) = -10 - 6 = -16 \quad \text{✓}
$$

All equations are satisfied.

---

### Step 4: Compute $ a + b $

$$
a + b = -3 + 2 = \boxed{-1}
$$

---

### Final Answer

$$
\boxed{-1}
$$

[Assistant]
Explanation:
Okay, so I need to find the values of 'a' and 'b' such that when I multiply the two quadratic expressions (3t² + 5t + a) and (4t² + bt - 2), the result is the given quartic polynomial 12t⁴ + 26t³ - 8t² - 16t + 6. Then, once I find 'a' and 'b', I have to add them together to get the answer. Alright, let me start by recalling how to multiply two polynomials.

First, I know that when you multiply two polynomials, you have to use the distributive property, which means each term in the first polynomial gets multiplied by each term in the second polynomial. So, let me write down the two polynomials:

First polynomial: 3t² + 5t + a

Second polynomial: 4t² + bt - 2

So, multiplying them together would give me:

(3t²)(4t²) + (3t²)(bt) + (3t²)(-2) + (5t)(4t²) + (5t)(bt) + (5t)(-2) + (a)(4t²) + (a)(bt) + (a)(-2)

Let me compute each term step by step.

Starting with the first term: 3t² multiplied by 4t². Multiplying the coefficients: 3*4 = 12. Then, multiplying the variables: t²*t² = t⁴. So, that term is 12t⁴.

Next term: 3t² multiplied by bt. Coefficients: 3*b = 3b. Variables: t²*t = t³. So, that term is 3b t³.

Third term: 3t² multiplied by -2. Coefficients: 3*(-2) = -6. Variables: t². So, that term is -6t².

Fourth term: 5t multiplied by 4t². Coefficients: 5*4 = 20. Variables: t*t² = t³. So, that term is 20t³.

Fifth term: 5t multiplied by bt. Coefficients: 5*b = 5b. Variables: t*t = t². So, that term is 5b t².

Sixth term: 5t multiplied by -2. Coefficients: 5*(-2) = -10. Variables: t. So, that term is -10t.

Seventh term: a multiplied by 4t². Coefficients: a*4 = 4a. Variables: t². So, that term is 4a t².

Eighth term: a multiplied by bt. Coefficients: a*b = ab. Variables: t. So, that term is ab t.

Ninth term: a multiplied by -2. Coefficients: a*(-2) = -2a. Variables: none, so just a constant term. So, that term is -2a.

Now, let me write all these terms together:

12t⁴ + 3b t³ - 6t² + 20t³ + 5b t² - 10t + 4a t² + ab t - 2a.

Now, I need to combine like terms. Let me group them by their degree:

- For t⁴: only 12t⁴.

- For t³: 3b t³ + 20t³. Let me factor out t³: (3b + 20) t³.

- For t²: -6t² + 5b t² + 4a t². Let me factor out t²: (-6 + 5b + 4a) t².

- For t: -10t + ab t. Factor out t: (-10 + ab) t.

- Constants: -2a.

So, putting it all together, the product is:

12t⁴ + (3b + 20) t³ + (-6 + 5b + 4a) t² + (-10 + ab) t - 2a.

Now, according to the problem, this product is equal to 12t⁴ + 26t³ - 8t² - 16t + 6. Therefore, the coefficients of corresponding terms must be equal. Let me set up equations for each coefficient.

Starting with the t⁴ term:

Coefficient of t⁴: 12 = 12. Well, that's already matching, so no new information here.

Next, the coefficient of t³: 3b + 20 = 26.

Then, the coefficient of t²: -6 + 5b + 4a = -8.

Coefficient of t: -10 + ab = -16.

Constant term: -2a = 6.

So, now I have four equations:

1) 3b + 20 = 26

2) -6 + 5b + 4a = -8

3) -10 + ab = -16

4) -2a = 6

Let me solve these equations step by step.

Starting with equation 4: -2a = 6. Solving for a:

Divide both sides by -2: a = 6 / (-2) = -3. So, a = -3.

Okay, so a is -3. Let me note that down: a = -3.

Now, let me plug this value of a into equation 2 and equation 3.

First, equation 2: -6 + 5b + 4a = -8.

Substituting a = -3:

-6 + 5b + 4*(-3) = -8

Compute 4*(-3) = -12.

So, equation becomes: -6 + 5b -12 = -8.

Combine constants: -6 -12 = -18.

So, -18 + 5b = -8.

Now, solving for b:

Add 18 to both sides: 5b = -8 + 18 = 10.

Therefore, 5b = 10 => b = 10 / 5 = 2. So, b = 2.

Alright, so b is 2. Let me check equation 1 to make sure.

Equation 1: 3b + 20 = 26.

Substituting b = 2: 3*2 + 20 = 6 + 20 = 26. Which matches. Good.

Now, let me check equation 3: -10 + ab = -16.

We have a = -3 and b = 2. So, ab = (-3)(2) = -6.

Therefore, -10 + (-6) = -16. Which is -16 = -16. Perfect, that checks out.

So, all equations are satisfied with a = -3 and b = 2. Therefore, the question asks for a + b. So, a + b = (-3) + 2 = -1.

Wait, but let me just make sure I didn't make any calculation errors. Let me verify the entire multiplication with these values of a and b to ensure that the product is indeed the given polynomial.

So, substituting a = -3 and b = 2 into the original polynomials:

First polynomial: 3t² + 5t + a = 3t² + 5t - 3.

Second polynomial: 4t² + bt - 2 = 4t² + 2t - 2.

Now, let me multiply these two polynomials step by step.

First, multiply 3t² by each term in the second polynomial:

3t² * 4t² = 12t⁴

3t² * 2t = 6t³

3t² * (-2) = -6t²

Next, multiply 5t by each term in the second polynomial:

5t * 4t² = 20t³

5t * 2t = 10t²

5t * (-2) = -10t

Then, multiply -3 by each term in the second polynomial:

-3 * 4t² = -12t²

-3 * 2t = -6t

-3 * (-2) = 6

Now, let me combine all these terms:

12t⁴ + 6t³ -6t² + 20t³ + 10t² -10t -12t² -6t + 6.

Now, combine like terms:

t⁴ term: 12t⁴

t³ terms: 6t³ + 20t³ = 26t³

t² terms: -6t² + 10t² -12t² = (-6 + 10 -12)t² = (-8)t²

t terms: -10t -6t = -16t

Constants: +6

So, altogether, the product is 12t⁴ + 26t³ -8t² -16t +6, which matches exactly the given polynomial. Perfect! So, my values for a and b are correct.

Therefore, a + b = -3 + 2 = -1.

But wait, the problem says "What is a + b?" So, the answer is -1? Let me check once again if I didn't mix up any signs.

Original equations:

From equation 4: -2a = 6 => a = -3. Correct.

Equation 1: 3b + 20 = 26 => 3b = 6 => b = 2. Correct.

Equation 2: -6 + 5b + 4a = -8. Plugging in a=-3, b=2:

-6 + 5*2 + 4*(-3) = -6 + 10 -12 = (-6 -12) +10 = -18 +10 = -8. Correct.

Equation 3: -10 + ab = -16. ab = (-3)(2) = -6. -10 + (-6) = -16. Correct.

All equations are satisfied. So, a + b = -3 + 2 = -1. Therefore, the answer is -1.

But let me check once again the multiplication step-by-step to be absolutely sure. Maybe I made a mistake in combining terms?

First polynomial: 3t² +5t -3

Second polynomial: 4t² +2t -2

Multiplying term by term:

First, 3t²*(4t² +2t -2) = 12t⁴ +6t³ -6t²

Then, 5t*(4t² +2t -2) = 20t³ +10t² -10t

Then, -3*(4t² +2t -2) = -12t² -6t +6

Adding all together:

12t⁴ +6t³ -6t² +20t³ +10t² -10t -12t² -6t +6

Combine t⁴: 12t⁴

t³: 6t³ +20t³ = 26t³

t²: -6t² +10t² -12t² = (-6 +10 -12)t² = (-8)t²

t terms: -10t -6t = -16t

Constants: +6

Yes, exactly matches. So, no mistake here. Therefore, the answer is indeed -1.

But just to make sure, let me check if there's another way to approach this problem, maybe by comparing coefficients differently or using another method. For example, maybe plugging in specific values of t to create equations? Although since we already have a consistent solution, maybe that's redundant, but just for thoroughness.

Alternatively, suppose I set t = 0. Let me try that. If t = 0, then the original product is:

(3*(0)^2 +5*(0) +a)*(4*(0)^2 +b*(0) -2) = (0 + 0 +a)*(0 + 0 -2) = a*(-2) = -2a.

On the other hand, the given polynomial evaluated at t=0 is 12*(0)^4 +26*(0)^3 -8*(0)^2 -16*(0) +6 = 0 + 0 -0 -0 +6 = 6.

Therefore, -2a = 6 => a = -3. Which matches our previous result. So that's a good check.

Similarly, maybe plug in t = 1? Let me try that.

Original product at t=1:

(3*(1)^2 +5*(1) +a)*(4*(1)^2 +b*(1) -2) = (3 +5 +a)*(4 +b -2) = (8 +a)*(2 +b)

Given polynomial at t=1: 12*(1)^4 +26*(1)^3 -8*(1)^2 -16*(1) +6 = 12 +26 -8 -16 +6 = Let's compute:

12 +26 = 38; 38 -8 = 30; 30 -16 = 14; 14 +6 = 20.

Therefore, (8 +a)(2 +b) = 20.

We already know that a = -3, so substituting:

(8 + (-3))*(2 + b) = (5)*(2 + b) = 20 => 5*(2 + b) = 20 => 2 + b = 4 => b = 2. Which again matches our previous result. So that's another check.

Alternatively, maybe plug in t = -1? Let me try that for thoroughness.

Original product at t=-1:

(3*(-1)^2 +5*(-1) +a)*(4*(-1)^2 +b*(-1) -2) = (3*1 -5 +a)*(4*1 -b -2) = (3 -5 +a)*(4 -b -2) = (-2 +a)*(2 -b)

Given polynomial at t=-1:

12*(-1)^4 +26*(-1)^3 -8*(-1)^2 -16*(-1) +6 = 12*1 +26*(-1) -8*1 -16*(-1) +6 = 12 -26 -8 +16 +6.

Calculating step by step:

12 -26 = -14; -14 -8 = -22; -22 +16 = -6; -6 +6 = 0.

Therefore, (-2 +a)*(2 -b) = 0.

We know a = -3 and b = 2, so substituting:

(-2 + (-3))*(2 - 2) = (-5)*(0) = 0. Which matches. So that's correct.

So, all these checks confirm that a = -3 and b = 2, so a + b = -1. Therefore, I can be confident that the answer is -1.

But just to make sure, let me check if there's any other possible solutions. For example, suppose there was a case where equations could have multiple solutions, but in this case, since we had four equations and two variables, but actually, the equations were not all independent. Let me check the number of equations.

We had four equations, but actually, the first equation (from t⁴ term) was redundant because it was already satisfied. Then, the other three equations (from t³, t², t, and constant term) gave us three equations. However, since we had two variables, a and b, we needed two equations. But in reality, we had four equations, but they were not all independent. Let me check:

Equation 1: 3b + 20 = 26 => 3b = 6 => b = 2.

Equation 4: -2a = 6 => a = -3.

Then equations 2 and 3 were dependent on these values. So, in reality, equations 1 and 4 gave us the values of a and b directly, and equations 2 and 3 were just checks. Therefore, there is a unique solution here. Therefore, there are no other solutions. So, the answer is definitely -1.

Therefore, I think I can conclude that a + b = -1.

**Final Answer**
The value of $a + b$ is \boxed{-1}.

To solve the problem, we are given two quadratic polynomials:

$$
(3t^2 + 5t + a)(4t^2 + bt - 2)
$$

and their product is:

$$
12t^4 + 26t^3 - 8t^2 - 16t + 6
$$

Our goal is to find the values of $ a $ and $ b $, and then compute $ a + b $.

---

### Step 1: Multiply the Polynomials

We expand the product:

$$
(3t^2 + 5t + a)(4t^2 + bt - 2)
$$

Using the distributive property, we compute each term:

- $ 3t^2 \cdot 4t^2 = 12t^4 $
- $ 3t^2 \cdot bt = 3bt^3 $
- $ 3t^2 \cdot (-2) = -6t^2 $
- $ 5t \cdot 4t^2 = 20t^3 $
- $ 5t \cdot bt = 5bt^2 $
- $ 5t \cdot (-2) = -10t $
- $ a \cdot 4t^2 = 4at^2 $
- $ a \cdot bt = abt $
- $ a \cdot (-2) = -2a $

Now, combine like terms:

- $ t^4 $: $ 12t^4 $
- $ t^3 $: $ (3b + 20)t^3 $
- $ t^2 $: $ (-6 + 5b + 4a)t^2 $
- $ t $: $ (-10 + ab)t $
- Constant: $ -2a $

So the product becomes:

$$
12t^4 + (3b + 20)t^3 + (-6 + 5b + 4a)t^2 + (-10 + ab)t - 2a
$$

---

### Step 2: Match Coefficients

We are told this equals:

$$
12t^4 + 26t^3 - 8t^2 - 16t + 6
$$

Now, equate the coefficients of corresponding powers of $ t $:

1. $ 12 = 12 $ (already matches)
2. $ 3b + 20 = 26 $ → $ 3b = 6 $ → $ b = 2 $
3. $ -6 + 5b + 4a = -8 $
4. $ -10 + ab = -16 $
5. $ -2a = 6 $ → $ a = -3 $

---

### Step 3: Solve for $ a $ and $ b $

From equation (5):
$$
-2a = 6 \Rightarrow a = -3
$$

Substitute $ a = -3 $ into equation (3):
$$
-6 + 5b + 4(-3) = -8 \Rightarrow -6 + 5b - 12 = -8 \Rightarrow 5b = 10 \Rightarrow b = 2
$$

Now verify with equation (4):
$$
-10 + ab = -10 + (-3)(2) = -10 - 6 = -16 \quad \text{✓}
$$

All equations are satisfied.

---

### Step 4: Compute $ a + b $

$$
a + b = -3 + 2 = \boxed{-1}
$$

---

### Final Answer

$$
\boxed{-1}
$$

Solution: The constant term of the product of two polynomials is just the product of the two constant terms. Therefore we know that $6=-2a$, so $a=-3$. We now consider the linear term of the product of our polynomials. It's given by $-16t=(5t\cdot-2)+a\cdot bt\Longrightarrow-16t=-10t+(-3)bt\Longrightarrow b=2$. Therefore our answer is $a+b=\boxed{-1}$.


Explanation:
Okay, so I need to find the radius of a right cylindrical tank. The problem says that the tank is being filled with water at a rate of 20π cubic meters per hour, and as the tank is filled, the water level rises four meters per hour. Hmm, let me think about how to approach this.

First, I remember that the volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height (or in this case, the water level). The problem mentions the rate at which the volume is increasing and the rate at which the height is increasing. So, this seems like a related rates problem from calculus. But wait, maybe I don't need calculus here? Let me check.

The problem states that the water is being filled at a rate of 20π cubic meters per hour. That's the rate of change of volume with respect to time, dV/dt. And the water level is rising at 4 meters per hour, which is the rate of change of height with respect to time, dh/dt. So, if I can relate these two rates using the volume formula, I can solve for the radius.

Let me write down what I know:

- dV/dt = 20π m³/hour
- dh/dt = 4 m/hour
- V = πr²h

Since the tank is a right cylinder, the radius is constant, so when I take the derivative of V with respect to time, I can use the product rule. Let me do that.

dV/dt = d/dt [πr²h] = πr² * dh/dt + π * 2r * dr/dt * h. Wait, but hold on, the radius is constant, right? The tank is a fixed cylinder, so dr/dt should be zero. Therefore, the derivative simplifies to dV/dt = πr² * dh/dt.

Wait, so if that's the case, then I can plug in the known values:

20π = πr² * 4

Then, I can divide both sides by π to get:

20 = 4r²

Then divide both sides by 4:

5 = r²

Therefore, r = √5. Is that it? Wait, that seems straightforward, but let me check again.

Hold on, maybe I made a mistake in the derivative. Let me verify. The volume is πr²h. If r is constant, then dV/dt is πr² * dh/dt. Yes, that's correct. So if I have dV/dt = πr² * dh/dt, then solving for r:

r² = (dV/dt) / (π * dh/dt) = (20π) / (π * 4) = 20 / 4 = 5. Therefore, r = √5. So the radius is √5 meters. That seems right.

But let me think again. Maybe there's a trick here? The problem says "a right cylindrical tank with circular bases is being filled with water at a rate of 20π cubic meters per hour. As the tank is filled, the water level rises four meters per hour." So, is there any chance that the tank isn't full? Like, maybe the rate of water level rising is dependent on the volume? But no, if the tank is a cylinder, then the cross-sectional area is constant, so the rate of change of volume is equal to the cross-sectional area times the rate of change of height. So, dV/dt = A * dh/dt, where A is πr². Therefore, solving for A, A = (dV/dt) / (dh/dt) = 20π / 4 = 5π. Then, since A = πr², πr² = 5π, so r² = 5, so r = √5. Yep, same answer. So that seems to check out.

Wait, but why does the problem mention "as the tank is filled"? Maybe implying that the tank isn't full? But even if the tank isn't full, the relationship between dV/dt and dh/dt still holds because the cross-sectional area is constant. So regardless of how full the tank is, as long as the radius is constant, the rate of change of volume is equal to the cross-sectional area times the rate of change of height. Therefore, even if the tank is partially filled, the calculation still works. Therefore, the answer is √5. So maybe that's the answer.

But let me check again. Let me do the units. dV/dt is in cubic meters per hour, dh/dt is in meters per hour. So if I divide dV/dt by dh/dt, I get (m³/hour) / (m/hour) = m², which is the unit of area. So that's consistent. Then, since A = πr², so πr² = 5π, so r² = 5. So radius is √5. So that seems correct.

Alternatively, maybe I can think of it in terms of time. Suppose we let t be time in hours. Then, the volume at time t is V(t) = πr²h(t). The rate of change of volume is dV/dt = πr² dh/dt. So if we have dV/dt = 20π, and dh/dt = 4, then πr² * 4 = 20π. So again, same equation. So solving for r² = 5. So radius is √5. So that's the answer.

Wait, but the problem says "Express your answer in simplest radical form." √5 is already in simplest radical form, so that's the answer. Therefore, I think that's correct.

But let me think again. Maybe I need to consider something else? For example, is the tank being filled from the bottom, so that the height is increasing? But that's already considered in the problem. The water level rising four meters per hour. So, the height is increasing at 4 m/hour. So, the cross-sectional area is πr², so the volume increase per hour is cross-sectional area times 4 m. So, πr² * 4 = 20π. So, πr² = 5π, so r² = 5. So, r = √5. So, yes, that's correct.

Alternatively, maybe the problem is trying to trick me into thinking that the tank is full? But even if the tank is full, the relationship still holds. The fact that it's being filled doesn't affect the cross-sectional area. So, regardless of the height, the rate of change of volume is equal to the cross-sectional area times the rate of change of height. Therefore, the answer is √5. So, I think that's correct.

Therefore, I think the radius is √5 meters. So, the answer is \boxed{\sqrt{5}}.

**Final Answer**
The radius of the tank is \boxed{\sqrt{5}} meters.

To determine the radius of the cylindrical tank, we begin by analyzing the relationship between the volume of water being added and the rate at which the water level rises.

---

### Step 1: Understand the Given Information

- The **rate of change of volume** is given as:
  $$
  \frac{dV}{dt} = 20\pi \text{ cubic meters per hour}
  $$
- The **rate of change of height** (water level) is:
  $$
  \frac{dh}{dt} = 4 \text{ meters per hour}
  $$
- The volume of a cylinder is:
  $$
  V = \pi r^2 h
  $$
  where $ r $ is the radius and $ h $ is the height of the water.

---

### Step 2: Differentiate the Volume Formula

Since the radius $ r $ is constant (the tank is a right cylinder), the derivative of the volume with respect to time is:
$$
\frac{dV}{dt} = \pi r^2 \cdot \frac{dh}{dt}
$$

Substitute the known values:
$$
20\pi = \pi r^2 \cdot 4
$$

---

### Step 3: Solve for the Radius

Divide both sides of the equation by $ \pi $:
$$
20 = 4r^2
$$

Divide both sides by 4:
$$
5 = r^2
$$

Take the square root of both sides:
$$
r = \sqrt{5}
$$

---

### Final Answer

The radius of the tank is:

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

[Assistant]
Explanation:
Okay, so I need to find the radius of a right cylindrical tank. The problem says that the tank is being filled with water at a rate of 20π cubic meters per hour, and as the tank is filled, the water level rises four meters per hour. Hmm, let me think about how to approach this.

First, I remember that the volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height (or in this case, the water level). The problem mentions the rate at which the volume is increasing and the rate at which the height is increasing. So, this seems like a related rates problem from calculus. But wait, maybe I don't need calculus here? Let me check.

The problem states that the water is being filled at a rate of 20π cubic meters per hour. That's the rate of change of volume with respect to time, dV/dt. And the water level is rising at 4 meters per hour, which is the rate of change of height with respect to time, dh/dt. So, if I can relate these two rates using the volume formula, I can solve for the radius.

Let me write down what I know:

- dV/dt = 20π m³/hour
- dh/dt = 4 m/hour
- V = πr²h

Since the tank is a right cylinder, the radius is constant, so when I take the derivative of V with respect to time, I can use the product rule. Let me do that.

dV/dt = d/dt [πr²h] = πr² * dh/dt + π * 2r * dr/dt * h. Wait, but hold on, the radius is constant, right? The tank is a fixed cylinder, so dr/dt should be zero. Therefore, the derivative simplifies to dV/dt = πr² * dh/dt.

Wait, so if that's the case, then I can plug in the known values:

20π = πr² * 4

Then, I can divide both sides by π to get:

20 = 4r²

Then divide both sides by 4:

5 = r²

Therefore, r = √5. Is that it? Wait, that seems straightforward, but let me check again.

Hold on, maybe I made a mistake in the derivative. Let me verify. The volume is πr²h. If r is constant, then dV/dt is πr² * dh/dt. Yes, that's correct. So if I have dV/dt = πr² * dh/dt, then solving for r:

r² = (dV/dt) / (π * dh/dt) = (20π) / (π * 4) = 20 / 4 = 5. Therefore, r = √5. So the radius is √5 meters. That seems right.

But let me think again. Maybe there's a trick here? The problem says "a right cylindrical tank with circular bases is being filled with water at a rate of 20π cubic meters per hour. As the tank is filled, the water level rises four meters per hour." So, is there any chance that the tank isn't full? Like, maybe the rate of water level rising is dependent on the volume? But no, if the tank is a cylinder, then the cross-sectional area is constant, so the rate of change of volume is equal to the cross-sectional area times the rate of change of height. So, dV/dt = A * dh/dt, where A is πr². Therefore, solving for A, A = (dV/dt) / (dh/dt) = 20π / 4 = 5π. Then, since A = πr², πr² = 5π, so r² = 5, so r = √5. Yep, same answer. So that seems to check out.

Wait, but why does the problem mention "as the tank is filled"? Maybe implying that the tank isn't full? But even if the tank isn't full, the relationship between dV/dt and dh/dt still holds because the cross-sectional area is constant. So regardless of how full the tank is, as long as the radius is constant, the rate of change of volume is equal to the cross-sectional area times the rate of change of height. Therefore, even if the tank is partially filled, the calculation still works. Therefore, the answer is √5. So maybe that's the answer.

But let me check again. Let me do the units. dV/dt is in cubic meters per hour, dh/dt is in meters per hour. So if I divide dV/dt by dh/dt, I get (m³/hour) / (m/hour) = m², which is the unit of area. So that's consistent. Then, since A = πr², so πr² = 5π, so r² = 5. So radius is √5. So that seems correct.

Alternatively, maybe I can think of it in terms of time. Suppose we let t be time in hours. Then, the volume at time t is V(t) = πr²h(t). The rate of change of volume is dV/dt = πr² dh/dt. So if we have dV/dt = 20π, and dh/dt = 4, then πr² * 4 = 20π. So again, same equation. So solving for r² = 5. So radius is √5. So that's the answer.

Wait, but the problem says "Express your answer in simplest radical form." √5 is already in simplest radical form, so that's the answer. Therefore, I think that's correct.

But let me think again. Maybe I need to consider something else? For example, is the tank being filled from the bottom, so that the height is increasing? But that's already considered in the problem. The water level rising four meters per hour. So, the height is increasing at 4 m/hour. So, the cross-sectional area is πr², so the volume increase per hour is cross-sectional area times 4 m. So, πr² * 4 = 20π. So, πr² = 5π, so r² = 5. So, r = √5. So, yes, that's correct.

Alternatively, maybe the problem is trying to trick me into thinking that the tank is full? But even if the tank is full, the relationship still holds. The fact that it's being filled doesn't affect the cross-sectional area. So, regardless of the height, the rate of change of volume is equal to the cross-sectional area times the rate of change of height. Therefore, the answer is √5. So, I think that's correct.

Therefore, I think the radius is √5 meters. So, the answer is \boxed{\sqrt{5}}.

**Final Answer**
The radius of the tank is \boxed{\sqrt{5}} meters.

To determine the radius of the cylindrical tank, we begin by analyzing the relationship between the volume of water being added and the rate at which the water level rises.

---

### Step 1: Understand the Given Information

- The **rate of change of volume** is given as:
  $$
  \frac{dV}{dt} = 20\pi \text{ cubic meters per hour}
  $$
- The **rate of change of height** (water level) is:
  $$
  \frac{dh}{dt} = 4 \text{ meters per hour}
  $$
- The volume of a cylinder is:
  $$
  V = \pi r^2 h
  $$
  where $ r $ is the radius and $ h $ is the height of the water.

---

### Step 2: Differentiate the Volume Formula

Since the radius $ r $ is constant (the tank is a right cylinder), the derivative of the volume with respect to time is:
$$
\frac{dV}{dt} = \pi r^2 \cdot \frac{dh}{dt}
$$

Substitute the known values:
$$
20\pi = \pi r^2 \cdot 4
$$

---

### Step 3: Solve for the Radius

Divide both sides of the equation by $ \pi $:
$$
20 = 4r^2
$$

Divide both sides by 4:
$$
5 = r^2
$$

Take the square root of both sides:
$$
r = \sqrt{5}
$$

---

### Final Answer

The radius of the tank is:

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

Solution:
The volume of the water increases by $20\pi$ cubic meters each hour, while the height of the water in the tank rises by 4 meters each hour. The volume of a right cylinder is $\pi r^2h$. If we look at the changes in volume and height in just one hour, we can solve for the radius.  \begin{align*}
\pi r^2h_f-\pi r^2h_0&=V_f-V_0\quad\Rightarrow\\
\pi r^2(\Delta h)&=\Delta V\quad\Rightarrow\\
\pi r^2(4)&=20\pi\quad\Rightarrow\\
4r^2&=20\quad\Rightarrow\\
r^2&=5
\end{align*} Since the radius must be positive, $r=\boxed{\sqrt{5}}$ meters.


Explanation:
Okay, so I need to find the greatest common divisor (GCD) of three numbers: 3339, 2961, and 1491. Hmm, let me remember how to do this. I think 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, 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 says that GCD(a, b) = GCD(b, a mod b), right? So I can use that to find the GCD of 3339 and 2961. Let me write that down.

Let me compute 3339 divided by 2961. Well, 2961 goes into 3339 once, with a remainder. Let me calculate the remainder: 3339 - 2961 = 378. So, GCD(3339, 2961) = GCD(2961, 378). Now I need to find GCD(2961, 378). Let me do the same step again.

Divide 2961 by 378. Let me see how many times 378 goes into 2961. Let me calculate 378 * 7 = 2646. Then 378 * 8 = 3024, which is too big because 3024 is larger than 2961. So it goes in 7 times with a remainder. Let me subtract 2646 from 2961: 2961 - 2646 = 315. So now, GCD(2961, 378) = GCD(378, 315).

Next step: GCD(378, 315). Let me divide 378 by 315. That goes in once, with a remainder of 378 - 315 = 63. So now it's GCD(315, 63).

Now, GCD(315, 63). Let me divide 315 by 63. 63 * 5 = 315, so the remainder is 0. That means the GCD is 63. So, GCD(3339, 2961) is 63.

Okay, so now I need to find the GCD of 63 and the third number, which is 1491. Let me do that. So GCD(63, 1491). Let me apply the Euclidean algorithm again.

Divide 1491 by 63. Let me see how many times 63 goes into 1491. Let me calculate 63 * 20 = 1260. Then 1491 - 1260 = 231. So the remainder is 231. Therefore, GCD(1491, 63) = GCD(63, 231).

Now, GCD(63, 231). Let me divide 231 by 63. 63 * 3 = 189. Subtract that from 231: 231 - 189 = 42. So now it's GCD(63, 42).

Next step: GCD(63, 42). Divide 63 by 42. That goes in once with a remainder of 63 - 42 = 21. So now it's GCD(42, 21).

Divide 42 by 21. That goes in exactly 2 times with remainder 0. Therefore, the GCD is 21.

So putting it all together, the GCD of 3339, 2961, and 1491 is 21. Wait, let me check if that's correct. Maybe I made a mistake somewhere. Let me verify each step again.

First, GCD(3339, 2961):

3339 divided by 2961 is 1 with remainder 378. Then GCD(2961, 378). 2961 divided by 378: 378*7=2646, 2961-2646=315. Then GCD(378, 315). 378 divided by 315 is 1 with remainder 63. Then GCD(315, 63). 315 divided by 63 is 5 with remainder 0. So yes, GCD is 63. Then GCD(63, 1491). 1491 divided by 63: 63*20=1260, 1491-1260=231. Then GCD(63, 231). 231 divided by 63 is 3 with remainder 42. Then GCD(63, 42). 63 divided by 42 is 1 with remainder 21. Then GCD(42, 21). 42 divided by 21 is 2 with remainder 0. So GCD is 21. That seems correct.

But let me check if 21 divides all three numbers. Let me check 3339 divided by 21. 21*100=2100, 3339-2100=1239. 21*50=1050, 1239-1050=189. 21*9=189. So total is 100+50+9=159. So 21*159=3339. Okay, that works.

Now 2961 divided by 21. Let me do 21*100=2100, 2961-2100=861. 21*40=840, 861-840=21. So that's 100+40+1=141. 21*141=2961. Correct.

Now 1491 divided by 21. Let me see. 21*70=1470. 1491-1470=21. So that's 70+1=71. 21*71=1491. Correct. So 21 divides all three numbers. Is there a larger number that divides all three? Let me check if 63 divides 1491. Wait, earlier steps said that GCD(63, 1491) is 21, so 63 does not divide 1491. Let me check. 1491 divided by 63. 63*20=1260, 1491-1260=231. 231 divided by 63 is 3.75? Wait, 63*3=189, 231-189=42. So 63*3 + 42 = 231. So 1491 divided by 63 is 23. So 63*23=1491? Wait, 63*20=1260, 63*3=189, so 1260+189=1449. But 1449 is less than 1491. 1491-1449=42. So 63*23=1449, remainder 42. Therefore, 63 does not divide 1491. Therefore, 63 is not a common divisor. Therefore, 21 is the GCD.

But just to be thorough, let me check if there is a higher common divisor. Let me check if 21 is the highest. Let me check if 21 is the GCD. Since we used the Euclidean algorithm, which is reliable, and verified that 21 divides all three numbers, and that there is no higher number, since the GCD of 63 and 1491 is 21, which is less than 63. Therefore, 21 is the GCD.

Alternatively, maybe I can factor all three numbers into primes and see. Let me try that approach as a cross-check.

Starting with 3339. Let me factor 3339. Let me check divisibility by small primes. First, check if it's even? No. Sum of digits: 3+3+3+9=18, which is divisible by 9, so 3339 is divisible by 9. Let me divide 3339 by 9. 9*371=3339? Let me check 9*300=2700, 9*70=630, 9*1=9. 2700+630=3330, plus 9 is 3339. Yes. So 3339=9*371. Now factor 371. Let me check divisibility. 371 divided by 7? 7*53=371? 7*50=350, 7*3=21, so 350+21=371. Yes! So 371=7*53. Therefore, 3339=9*7*53. Which is 3^2 * 7 * 53.

Now 2961. Let me factor 2961. Again, check divisibility by 3. Sum of digits: 2+9+6+1=18, which is divisible by 9. So 2961 is divisible by 9. Let me divide 2961 by 9. 9*300=2700, 2961-2700=261. 261 divided by 9 is 29. So total is 300 + 29 = 329. Therefore, 2961=9*329. Now factor 329. Let me check divisibility. 329 divided by 7? 7*47=329? 7*40=280, 7*7=49, so 280+49=329. Yes! So 329=7*47. Therefore, 2961=9*7*47=3^2 *7*47.

Now 1491. Let me factor 1491. Check divisibility by 3: 1+4+9+1=15, which is divisible by 3. So divide by 3: 1491 / 3 = 497. Now factor 497. Let me check divisibility. 497 divided by 7? 7*71=497? 7*70=490, plus 7 is 497. Yes! So 497=7*71. Therefore, 1491=3*7*71.

So now, the prime factors:

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

So the common prime factors among all three numbers are 3 and 7. The minimum exponents for these primes:

- For 3: the exponents are 2, 2, and 1. So minimum is 1.
- For 7: exponents are 1, 1, 1. So minimum is 1.

Therefore, GCD is 3^1 * 7^1 = 3*7=21. Which matches the previous result. So that confirms that the GCD is indeed 21.

Therefore, I can be confident that the greatest common divisor of 3339, 2961, and 1491 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 three numbers $ 3339 $, $ 2961 $, and $ 1491 $, we can use the **Euclidean algorithm**, which is a reliable method for computing the GCD of two numbers. Since we are dealing with three numbers, we can apply the algorithm iteratively: first compute the GCD of the first two numbers, and then compute the GCD of that result with the third number.

---

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

We apply the Euclidean algorithm:

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

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

---

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

Now we compute the GCD of the result from the previous step (63) with the third number (1491):

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

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

---

### Step 3: Final Result

Therefore, the GCD of all three numbers is:

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

---

### Verification via Prime Factorization

To double-check, we can factor each number into its prime components:

- $ 3339 = 3^2 \times 7 \times 53 $
- $ 2961 = 3^2 \times 7 \times 47 $
- $ 1491 = 3 \times 7 \times 71 $

The common prime factors among all three are $ 3 $ and $ 7 $, with the smallest exponents being $ 3^1 $ and $ 7^1 $. Thus:

$$
\gcd = 3^1 \times 7^1 = 21
$$

---

### Final Answer

$$
\boxed{21}
$$

[Assistant]
Explanation:
Okay, so I need to find the greatest common divisor (GCD) of three numbers: 3339, 2961, and 1491. Hmm, let me remember how to do this. I think 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, 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 says that GCD(a, b) = GCD(b, a mod b), right? So I can use that to find the GCD of 3339 and 2961. Let me write that down.

Let me compute 3339 divided by 2961. Well, 2961 goes into 3339 once, with a remainder. Let me calculate the remainder: 3339 - 2961 = 378. So, GCD(3339, 2961) = GCD(2961, 378). Now I need to find GCD(2961, 378). Let me do the same step again.

Divide 2961 by 378. Let me see how many times 378 goes into 2961. Let me calculate 378 * 7 = 2646. Then 378 * 8 = 3024, which is too big because 3024 is larger than 2961. So it goes in 7 times with a remainder. Let me subtract 2646 from 2961: 2961 - 2646 = 315. So now, GCD(2961, 378) = GCD(378, 315).

Next step: GCD(378, 315). Let me divide 378 by 315. That goes in once, with a remainder of 378 - 315 = 63. So now it's GCD(315, 63).

Now, GCD(315, 63). Let me divide 315 by 63. 63 * 5 = 315, so the remainder is 0. That means the GCD is 63. So, GCD(3339, 2961) is 63.

Okay, so now I need to find the GCD of 63 and the third number, which is 1491. Let me do that. So GCD(63, 1491). Let me apply the Euclidean algorithm again.

Divide 1491 by 63. Let me see how many times 63 goes into 1491. Let me calculate 63 * 20 = 1260. Then 1491 - 1260 = 231. So the remainder is 231. Therefore, GCD(1491, 63) = GCD(63, 231).

Now, GCD(63, 231). Let me divide 231 by 63. 63 * 3 = 189. Subtract that from 231: 231 - 189 = 42. So now it's GCD(63, 42).

Next step: GCD(63, 42). Divide 63 by 42. That goes in once with a remainder of 63 - 42 = 21. So now it's GCD(42, 21).

Divide 42 by 21. That goes in exactly 2 times with remainder 0. Therefore, the GCD is 21.

So putting it all together, the GCD of 3339, 2961, and 1491 is 21. Wait, let me check if that's correct. Maybe I made a mistake somewhere. Let me verify each step again.

First, GCD(3339, 2961):

3339 divided by 2961 is 1 with remainder 378. Then GCD(2961, 378). 2961 divided by 378: 378*7=2646, 2961-2646=315. Then GCD(378, 315). 378 divided by 315 is 1 with remainder 63. Then GCD(315, 63). 315 divided by 63 is 5 with remainder 0. So yes, GCD is 63. Then GCD(63, 1491). 1491 divided by 63: 63*20=1260, 1491-1260=231. Then GCD(63, 231). 231 divided by 63 is 3 with remainder 42. Then GCD(63, 42). 63 divided by 42 is 1 with remainder 21. Then GCD(42, 21). 42 divided by 21 is 2 with remainder 0. So GCD is 21. That seems correct.

But let me check if 21 divides all three numbers. Let me check 3339 divided by 21. 21*100=2100, 3339-2100=1239. 21*50=1050, 1239-1050=189. 21*9=189. So total is 100+50+9=159. So 21*159=3339. Okay, that works.

Now 2961 divided by 21. Let me do 21*100=2100, 2961-2100=861. 21*40=840, 861-840=21. So that's 100+40+1=141. 21*141=2961. Correct.

Now 1491 divided by 21. Let me see. 21*70=1470. 1491-1470=21. So that's 70+1=71. 21*71=1491. Correct. So 21 divides all three numbers. Is there a larger number that divides all three? Let me check if 63 divides 1491. Wait, earlier steps said that GCD(63, 1491) is 21, so 63 does not divide 1491. Let me check. 1491 divided by 63. 63*20=1260, 1491-1260=231. 231 divided by 63 is 3.75? Wait, 63*3=189, 231-189=42. So 63*3 + 42 = 231. So 1491 divided by 63 is 23. So 63*23=1491? Wait, 63*20=1260, 63*3=189, so 1260+189=1449. But 1449 is less than 1491. 1491-1449=42. So 63*23=1449, remainder 42. Therefore, 63 does not divide 1491. Therefore, 63 is not a common divisor. Therefore, 21 is the GCD.

But just to be thorough, let me check if there is a higher common divisor. Let me check if 21 is the highest. Let me check if 21 is the GCD. Since we used the Euclidean algorithm, which is reliable, and verified that 21 divides all three numbers, and that there is no higher number, since the GCD of 63 and 1491 is 21, which is less than 63. Therefore, 21 is the GCD.

Alternatively, maybe I can factor all three numbers into primes and see. Let me try that approach as a cross-check.

Starting with 3339. Let me factor 3339. Let me check divisibility by small primes. First, check if it's even? No. Sum of digits: 3+3+3+9=18, which is divisible by 9, so 3339 is divisible by 9. Let me divide 3339 by 9. 9*371=3339? Let me check 9*300=2700, 9*70=630, 9*1=9. 2700+630=3330, plus 9 is 3339. Yes. So 3339=9*371. Now factor 371. Let me check divisibility. 371 divided by 7? 7*53=371? 7*50=350, 7*3=21, so 350+21=371. Yes! So 371=7*53. Therefore, 3339=9*7*53. Which is 3^2 * 7 * 53.

Now 2961. Let me factor 2961. Again, check divisibility by 3. Sum of digits: 2+9+6+1=18, which is divisible by 9. So 2961 is divisible by 9. Let me divide 2961 by 9. 9*300=2700, 2961-2700=261. 261 divided by 9 is 29. So total is 300 + 29 = 329. Therefore, 2961=9*329. Now factor 329. Let me check divisibility. 329 divided by 7? 7*47=329? 7*40=280, 7*7=49, so 280+49=329. Yes! So 329=7*47. Therefore, 2961=9*7*47=3^2 *7*47.

Now 1491. Let me factor 1491. Check divisibility by 3: 1+4+9+1=15, which is divisible by 3. So divide by 3: 1491 / 3 = 497. Now factor 497. Let me check divisibility. 497 divided by 7? 7*71=497? 7*70=490, plus 7 is 497. Yes! So 497=7*71. Therefore, 1491=3*7*71.

So now, the prime factors:

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

So the common prime factors among all three numbers are 3 and 7. The minimum exponents for these primes:

- For 3: the exponents are 2, 2, and 1. So minimum is 1.
- For 7: exponents are 1, 1, 1. So minimum is 1.

Therefore, GCD is 3^1 * 7^1 = 3*7=21. Which matches the previous result. So that confirms that the GCD is indeed 21.

Therefore, I can be confident that the greatest common divisor of 3339, 2961, and 1491 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 three numbers $ 3339 $, $ 2961 $, and $ 1491 $, we can use the **Euclidean algorithm**, which is a reliable method for computing the GCD of two numbers. Since we are dealing with three numbers, we can apply the algorithm iteratively: first compute the GCD of the first two numbers, and then compute the GCD of that result with the third number.

---

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

We apply the Euclidean algorithm:

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

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

---

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

Now we compute the GCD of the result from the previous step (63) with the third number (1491):

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

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

---

### Step 3: Final Result

Therefore, the GCD of all three numbers is:

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

---

### Verification via Prime Factorization

To double-check, we can factor each number into its prime components:

- $ 3339 = 3^2 \times 7 \times 53 $
- $ 2961 = 3^2 \times 7 \times 47 $
- $ 1491 = 3 \times 7 \times 71 $

The common prime factors among all three are $ 3 $ and $ 7 $, with the smallest exponents being $ 3^1 $ and $ 7^1 $. Thus:

$$
\gcd = 3^1 \times 7^1 = 21
$$

---

### Final Answer

$$
\boxed{21}
$$

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 who's 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, we can use the binomial probability formula. Is this a binomial situation? Let me check the conditions. There are a fixed number of trials, which here are the seven islands. Each trial (island) has two possible outcomes: treasure or no treasure. The probability of success (finding treasure) is constant at 1/5 for each island. And the trials are independent because the presence of treasure on one island doesn't affect the others. So yes, this is a binomial distribution problem.

The formula for binomial probability is P(k) = C(n, k) * p^k * (1 - p)^(n - k), where:
- P(k) is the probability of exactly k successes,
- C(n, k) is the combination of n things taken k at a time,
- p is the probability of success on a single trial,
- n is the number of trials.

In this case, n = 7, k = 4, p = 1/5. So plugging these into the formula, we get:

P(4) = C(7, 4) * (1/5)^4 * (4/5)^(7 - 4)

Let me compute each part step by step.

First, calculate C(7, 4). The combination formula is C(n, k) = n! / (k! * (n - k)!).

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

Calculating factorials:
7! = 5040
4! = 24
3! = 6

So C(7, 4) = 5040 / (24 * 6) = 5040 / 144. Let me compute that. Dividing 5040 by 144. Let's see, 144 * 35 = 5040 because 144 * 30 = 4320, and 144 * 5 = 720, so 4320 + 720 = 5040. So C(7, 4) = 35.

Okay, so that part is 35.

Next, (1/5)^4. Let me compute that. (1/5)^4 = 1 / (5^4) = 1 / 625.

Then, (4/5)^(7 - 4) = (4/5)^3. Let me calculate that. (4/5)^3 = 64 / 125.

So putting it all together:

P(4) = 35 * (1/625) * (64/125)

Now, let me compute this step by step. First, multiply 35 and 1/625. 35 / 625. Let me simplify that fraction. Both numerator and denominator are divisible by 5: 35 ÷ 5 = 7, 625 ÷ 5 = 125. So that becomes 7/125.

Then, multiply that by 64/125. So 7/125 * 64/125. Multiply numerators: 7 * 64 = 448. Multiply denominators: 125 * 125 = 15625. So the probability is 448 / 15625.

Wait, let me check if that can be simplified. Let's see if 448 and 15625 have any common factors. 448 factors into 2^6 * 7 (since 448 ÷ 2 = 224, ÷2=112, ÷2=56, ÷2=28, ÷2=14, ÷2=7). 15625 is 5^6. There are no common prime factors between numerator and denominator, so 448/15625 is the simplified fraction.

Alternatively, if I want to write it as a decimal, I can divide 448 by 15625. Let me do that. Let me see:

First, note that 15625 * 0.028 = 15625 * 0.02 + 15625 * 0.008 = 312.5 + 125 = 437.5. Hmm, 0.028 gives 437.5. The numerator is 448, which is 448 - 437.5 = 10.5 more. So 10.5 / 15625 = 0.000672. So total is approximately 0.028 + 0.000672 = 0.028672. So approximately 2.8672%.

But the question doesn't specify the form of the answer, so maybe they want the exact fraction. Let me check if I did all the calculations correctly.

Wait, let me verify the combination again. C(7,4) is 35. Yes, because C(7,4) is same as C(7,3) which is 35. Let me confirm with another method. For example, 7 choose 4: 7*6*5*4 / 4*3*2*1 = (7*6*5*4)/24. Let me compute that. 7*6=42, 42*5=210, 210*4=840. 840 divided by 24 is 35. Yes, that's correct.

Then (1/5)^4 is 1/625, and (4/5)^3 is 64/125. Multiplying them together with 35: 35*(1/625)*(64/125). Let me check the multiplication again. 35*64 = 2240, and then divided by 625*125 = 78125? Wait, wait, no. Wait, 35*(1/625)*(64/125) is equal to (35*64)/(625*125). Let me compute denominator: 625*125. 625 is 5^4, 125 is 5^3, so 5^7 = 78125. Numerator: 35*64. Let me compute 35*64. 35*60=2100, 35*4=140, so total is 2100+140=2240. So 2240 / 78125. Wait, earlier I had 448 / 15625. Wait, 2240 divided by 5 is 448, and 78125 divided by 5 is 15625. So yes, 2240/78125 = 448/15625. So that's correct.

So the exact probability is 448/15625. Let me check if that reduces further. As I thought before, 448 is 2^6 * 7, and 15625 is 5^6. No common factors, so that's the simplified fraction. Alternatively, if they want a decimal, approximately 0.028672, which is about 2.87%.

But let me check if I made a mistake in the exponents. Wait, the formula is C(n, k) * p^k * (1 - p)^{n - k}. So n is 7, k is 4, so (1 - p) is 4/5, and the exponent is 7 - 4 = 3. So that's correct. So (4/5)^3. So that's correct. So the calculation seems right.

Alternatively, maybe I can think of it as 35 * (1/5)^4 * (4/5)^3. Let me compute that again. 35*(1/5)^4*(4/5)^3. Let me write all terms with denominator 5^7. Because (1/5)^4 is 1/5^4 and (4/5)^3 is 4^3 / 5^3. So multiplying them together, we have 1*4^3 / 5^(4+3) = 64 / 5^7. Then multiplied by 35. So total is 35*64 / 5^7. Let me compute 5^7. 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125. So yes, denominator is 78125. Numerator is 35*64. Which is 2240. So 2240/78125. Which reduces to 448/15625. Yep, same as before.

So that seems correct. Therefore, the probability is 448/15625. If I want to write that as a decimal, as I did before, approximately 0.028672, which is about 2.87%. But maybe the question expects the fractional form. Let me check if 448/15625 can be simplified further. Let me check GCD of 448 and 15625. The prime factors of 448 are 2^6 * 7. The prime factors of 15625 are 5^6. There are no common prime factors, so yes, it's reduced completely.

Alternatively, maybe I can write it as a decimal or a percentage, but unless specified, the fraction is probably the better answer. Let me check if 448 divided by 15625 is indeed the correct value. Let me do a quick check with another approach.

Alternatively, think of the probability as the number of ways to choose 4 islands out of 7, times the probability that those 4 have treasure and the remaining 3 don't. So that's exactly what we did. So 35 * (1/5)^4 * (4/5)^3. So that's correct.

Alternatively, maybe I can use decimal approximations step by step to verify. Let me compute each part numerically:

C(7,4) = 35.

(1/5)^4 = 0.0016.

(4/5)^3 = (0.8)^3 = 0.512.

Multiply them all together: 35 * 0.0016 * 0.512.

First, 35 * 0.0016. Let me compute 35 * 0.0016. 35 * 0.001 = 0.035, 35 * 0.0006 = 0.021. So total is 0.035 + 0.021 = 0.056.

Then multiply by 0.512: 0.056 * 0.512. Let me compute that. 0.05 * 0.512 = 0.0256, and 0.006 * 0.512 = 0.003072. Adding them together: 0.0256 + 0.003072 = 0.028672. Which matches the decimal I had earlier. So 0.028672, which is 448/15625. So that seems correct.

Therefore, the probability is 448/15625. If I wanted to write this as a reduced fraction, but since it's already reduced, that's the answer. Alternatively, if they want it as a decimal, approximately 0.0287, but the question doesn't specify. Since the original probabilities are given as fractions, it's safer to present the exact fraction.

Wait, but let me check once more if I didn't make a mistake in the combination. For example, if I thought C(7,4) was something else. Let me recall that C(7,4) is 35. Let me think of another way. For example, C(7,4) = C(6,3) + C(6,4). Wait, but maybe that's complicating. Alternatively, think of the number of ways to choose 4 islands out of 7. For example, the first island can be chosen in 7 ways, the second in 6, third in 5, fourth in 4, but since order doesn't matter, divide by 4!. So 7*6*5*4 / 4! = 7*6*5*4 / 24. Which is 7*6*5*4 / 24. Let me compute that. 7*6 is 42, 42*5 is 210, 210*4 is 840. 840 divided by 24 is 35. Yep, same as before. So that's correct.

Therefore, I think my answer is correct. So the probability is 448/15625. Let me check if that fraction can be simplified further. Let me check if 448 and 15625 have any common divisors. 448 is even, 15625 is odd, so 2 is not a common factor. 448 divided by 7 is 64, and 15625 divided by 7? Let me check 7*2232 = 15624, so 15625 divided by 7 is 2232.142..., so not an integer. Therefore, 7 is not a common factor. Therefore, the fraction is indeed reduced completely.

Alternatively, maybe I can write it as a decimal with more precision, but unless asked, the fraction is better. So I think that's the answer.

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

To determine the probability that exactly 4 out of 7 islands have treasure, we model this as a **binomial probability** problem. Here's the step-by-step breakdown:

---

### **Step 1: Identify the Parameters**

- **Number of trials (n):** 7 (since there are 7 islands)
- **Number of successes (k):** 4 (we want exactly 4 islands with treasure)
- **Probability of success (p):** $ \frac{1}{5} $ (chance of treasure on a single island)
- **Probability of failure (q):** $ 1 - p = \frac{4}{5} $

---

### **Step 2: Use the Binomial Probability Formula**

The binomial probability formula is:

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

Substituting the known values:

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

---

### **Step 3: Compute Each Component**

- **Combination:**
  $$
  \binom{7}{4} = \frac{7!}{4!(7 - 4)!} = \frac{7 \cdot 6 \cdot 5 \cdot 4}{4 \cdot 3 \cdot 2 \cdot 1} = 35
  $$

- **Probability of 4 successes:**
  $$
  \left(\frac{1}{5}\right)^4 = \frac{1}{625}
  $$

- **Probability of 3 failures:**
  $$
  \left(\frac{4}{5}\right)^3 = \frac{64}{125}
  $$

---

### **Step 4: Multiply All Components**

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

First, multiply the numerators:

$$
35 \cdot 64 = 2240
$$

Then, multiply the denominators:

$$
625 \cdot 125 = 78125
$$

So:

$$
P(4) = \frac{2240}{78125}
$$

Simplify the fraction by dividing numerator and denominator by 5:

$$
\frac{2240}{78125} = \frac{448}{15625}
$$

---

### **Final Answer**

The probability that exactly 4 of the islands have treasure is:

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

[Assistant]
Explanation:
Okay, so there's this pirate who's 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, we can use the binomial probability formula. Is this a binomial situation? Let me check the conditions. There are a fixed number of trials, which here are the seven islands. Each trial (island) has two possible outcomes: treasure or no treasure. The probability of success (finding treasure) is constant at 1/5 for each island. And the trials are independent because the presence of treasure on one island doesn't affect the others. So yes, this is a binomial distribution problem.

The formula for binomial probability is P(k) = C(n, k) * p^k * (1 - p)^(n - k), where:
- P(k) is the probability of exactly k successes,
- C(n, k) is the combination of n things taken k at a time,
- p is the probability of success on a single trial,
- n is the number of trials.

In this case, n = 7, k = 4, p = 1/5. So plugging these into the formula, we get:

P(4) = C(7, 4) * (1/5)^4 * (4/5)^(7 - 4)

Let me compute each part step by step.

First, calculate C(7, 4). The combination formula is C(n, k) = n! / (k! * (n - k)!).

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

Calculating factorials:
7! = 5040
4! = 24
3! = 6

So C(7, 4) = 5040 / (24 * 6) = 5040 / 144. Let me compute that. Dividing 5040 by 144. Let's see, 144 * 35 = 5040 because 144 * 30 = 4320, and 144 * 5 = 720, so 4320 + 720 = 5040. So C(7, 4) = 35.

Okay, so that part is 35.

Next, (1/5)^4. Let me compute that. (1/5)^4 = 1 / (5^4) = 1 / 625.

Then, (4/5)^(7 - 4) = (4/5)^3. Let me calculate that. (4/5)^3 = 64 / 125.

So putting it all together:

P(4) = 35 * (1/625) * (64/125)

Now, let me compute this step by step. First, multiply 35 and 1/625. 35 / 625. Let me simplify that fraction. Both numerator and denominator are divisible by 5: 35 ÷ 5 = 7, 625 ÷ 5 = 125. So that becomes 7/125.

Then, multiply that by 64/125. So 7/125 * 64/125. Multiply numerators: 7 * 64 = 448. Multiply denominators: 125 * 125 = 15625. So the probability is 448 / 15625.

Wait, let me check if that can be simplified. Let's see if 448 and 15625 have any common factors. 448 factors into 2^6 * 7 (since 448 ÷ 2 = 224, ÷2=112, ÷2=56, ÷2=28, ÷2=14, ÷2=7). 15625 is 5^6. There are no common prime factors between numerator and denominator, so 448/15625 is the simplified fraction.

Alternatively, if I want to write it as a decimal, I can divide 448 by 15625. Let me do that. Let me see:

First, note that 15625 * 0.028 = 15625 * 0.02 + 15625 * 0.008 = 312.5 + 125 = 437.5. Hmm, 0.028 gives 437.5. The numerator is 448, which is 448 - 437.5 = 10.5 more. So 10.5 / 15625 = 0.000672. So total is approximately 0.028 + 0.000672 = 0.028672. So approximately 2.8672%.

But the question doesn't specify the form of the answer, so maybe they want the exact fraction. Let me check if I did all the calculations correctly.

Wait, let me verify the combination again. C(7,4) is 35. Yes, because C(7,4) is same as C(7,3) which is 35. Let me confirm with another method. For example, 7 choose 4: 7*6*5*4 / 4*3*2*1 = (7*6*5*4)/24. Let me compute that. 7*6=42, 42*5=210, 210*4=840. 840 divided by 24 is 35. Yes, that's correct.

Then (1/5)^4 is 1/625, and (4/5)^3 is 64/125. Multiplying them together with 35: 35*(1/625)*(64/125). Let me check the multiplication again. 35*64 = 2240, and then divided by 625*125 = 78125? Wait, wait, no. Wait, 35*(1/625)*(64/125) is equal to (35*64)/(625*125). Let me compute denominator: 625*125. 625 is 5^4, 125 is 5^3, so 5^7 = 78125. Numerator: 35*64. Let me compute 35*64. 35*60=2100, 35*4=140, so total is 2100+140=2240. So 2240 / 78125. Wait, earlier I had 448 / 15625. Wait, 2240 divided by 5 is 448, and 78125 divided by 5 is 15625. So yes, 2240/78125 = 448/15625. So that's correct.

So the exact probability is 448/15625. Let me check if that reduces further. As I thought before, 448 is 2^6 * 7, and 15625 is 5^6. No common factors, so that's the simplified fraction. Alternatively, if they want a decimal, approximately 0.028672, which is about 2.87%.

But let me check if I made a mistake in the exponents. Wait, the formula is C(n, k) * p^k * (1 - p)^{n - k}. So n is 7, k is 4, so (1 - p) is 4/5, and the exponent is 7 - 4 = 3. So that's correct. So (4/5)^3. So that's correct. So the calculation seems right.

Alternatively, maybe I can think of it as 35 * (1/5)^4 * (4/5)^3. Let me compute that again. 35*(1/5)^4*(4/5)^3. Let me write all terms with denominator 5^7. Because (1/5)^4 is 1/5^4 and (4/5)^3 is 4^3 / 5^3. So multiplying them together, we have 1*4^3 / 5^(4+3) = 64 / 5^7. Then multiplied by 35. So total is 35*64 / 5^7. Let me compute 5^7. 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125. So yes, denominator is 78125. Numerator is 35*64. Which is 2240. So 2240/78125. Which reduces to 448/15625. Yep, same as before.

So that seems correct. Therefore, the probability is 448/15625. If I want to write that as a decimal, as I did before, approximately 0.028672, which is about 2.87%. But maybe the question expects the fractional form. Let me check if 448/15625 can be simplified further. Let me check GCD of 448 and 15625. The prime factors of 448 are 2^6 * 7. The prime factors of 15625 are 5^6. There are no common prime factors, so yes, it's reduced completely.

Alternatively, maybe I can write it as a decimal or a percentage, but unless specified, the fraction is probably the better answer. Let me check if 448 divided by 15625 is indeed the correct value. Let me do a quick check with another approach.

Alternatively, think of the probability as the number of ways to choose 4 islands out of 7, times the probability that those 4 have treasure and the remaining 3 don't. So that's exactly what we did. So 35 * (1/5)^4 * (4/5)^3. So that's correct.

Alternatively, maybe I can use decimal approximations step by step to verify. Let me compute each part numerically:

C(7,4) = 35.

(1/5)^4 = 0.0016.

(4/5)^3 = (0.8)^3 = 0.512.

Multiply them all together: 35 * 0.0016 * 0.512.

First, 35 * 0.0016. Let me compute 35 * 0.0016. 35 * 0.001 = 0.035, 35 * 0.0006 = 0.021. So total is 0.035 + 0.021 = 0.056.

Then multiply by 0.512: 0.056 * 0.512. Let me compute that. 0.05 * 0.512 = 0.0256, and 0.006 * 0.512 = 0.003072. Adding them together: 0.0256 + 0.003072 = 0.028672. Which matches the decimal I had earlier. So 0.028672, which is 448/15625. So that seems correct.

Therefore, the probability is 448/15625. If I wanted to write this as a reduced fraction, but since it's already reduced, that's the answer. Alternatively, if they want it as a decimal, approximately 0.0287, but the question doesn't specify. Since the original probabilities are given as fractions, it's safer to present the exact fraction.

Wait, but let me check once more if I didn't make a mistake in the combination. For example, if I thought C(7,4) was something else. Let me recall that C(7,4) is 35. Let me think of another way. For example, C(7,4) = C(6,3) + C(6,4). Wait, but maybe that's complicating. Alternatively, think of the number of ways to choose 4 islands out of 7. For example, the first island can be chosen in 7 ways, the second in 6, third in 5, fourth in 4, but since order doesn't matter, divide by 4!. So 7*6*5*4 / 4! = 7*6*5*4 / 24. Which is 7*6*5*4 / 24. Let me compute that. 7*6 is 42, 42*5 is 210, 210*4 is 840. 840 divided by 24 is 35. Yep, same as before. So that's correct.

Therefore, I think my answer is correct. So the probability is 448/15625. Let me check if that fraction can be simplified further. Let me check if 448 and 15625 have any common divisors. 448 is even, 15625 is odd, so 2 is not a common factor. 448 divided by 7 is 64, and 15625 divided by 7? Let me check 7*2232 = 15624, so 15625 divided by 7 is 2232.142..., so not an integer. Therefore, 7 is not a common factor. Therefore, the fraction is indeed reduced completely.

Alternatively, maybe I can write it as a decimal with more precision, but unless asked, the fraction is better. So I think that's the answer.

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

To determine the probability that exactly 4 out of 7 islands have treasure, we model this as a **binomial probability** problem. Here's the step-by-step breakdown:

---

### **Step 1: Identify the Parameters**

- **Number of trials (n):** 7 (since there are 7 islands)
- **Number of successes (k):** 4 (we want exactly 4 islands with treasure)
- **Probability of success (p):** $ \frac{1}{5} $ (chance of treasure on a single island)
- **Probability of failure (q):** $ 1 - p = \frac{4}{5} $

---

### **Step 2: Use the Binomial Probability Formula**

The binomial probability formula is:

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

Substituting the known values:

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

---

### **Step 3: Compute Each Component**

- **Combination:**
  $$
  \binom{7}{4} = \frac{7!}{4!(7 - 4)!} = \frac{7 \cdot 6 \cdot 5 \cdot 4}{4 \cdot 3 \cdot 2 \cdot 1} = 35
  $$

- **Probability of 4 successes:**
  $$
  \left(\frac{1}{5}\right)^4 = \frac{1}{625}
  $$

- **Probability of 3 failures:**
  $$
  \left(\frac{4}{5}\right)^3 = \frac{64}{125}
  $$

---

### **Step 4: Multiply All Components**

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

First, multiply the numerators:

$$
35 \cdot 64 = 2240
$$

Then, multiply the denominators:

$$
625 \cdot 125 = 78125
$$

So:

$$
P(4) = \frac{2240}{78125}
$$

Simplify the fraction by dividing numerator and denominator by 5:

$$
\frac{2240}{78125} = \frac{448}{15625}
$$

---

### **Final Answer**

The probability that exactly 4 of the islands have treasure is:

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

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 $ such that the circle given by the equation $ x^2 - 10x + y^2 + 6y + c = 0 $ has a radius of length 1. Hmm, let me recall how to approach this.

First, I remember that the general equation of a circle is $ (x - h)^2 + (y - k)^2 = r^2 $, where $ (h, k) $ is the center and $ r $ is the radius. The given equation isn't in that standard form, so I probably need to complete the square for both the $ x $ and $ y $ terms. That should help me rewrite the equation into the standard form and then compare it to find the value of $ c $.

Let me start by rearranging the given equation. The original equation is:

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

I need to group the $ x $ terms and the $ y $ terms together. Let me rewrite it as:

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

Now, I need to complete the square for each of the quadratic terms. Let me handle the $ x $ terms first: $ x^2 - 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 can write:

$ x^2 - 10x = (x - 5)^2 - 25 $

Wait, let me check that. If I expand $ (x - 5)^2 $, it's $ x^2 - 10x + 25 $, so to get back to $ x^2 - 10x $, I need to subtract 25. Yes, that's correct. So, $ x^2 - 10x = (x - 5)^2 - 25 $.

Now, moving on to the $ y $ terms: $ y^2 + 6y $. Similarly, take the coefficient of $ y $, which is 6, divide by 2 to get 3, square it to get 9. Therefore:

$ y^2 + 6y = (y + 3)^2 - 9 $

Let me verify that. Expanding $ (y + 3)^2 $ gives $ y^2 + 6y + 9 $, so subtracting 9 gives back $ y^2 + 6y $. Correct.

So substituting these back into the original equation:

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

Now, let me combine the constants:

-25 - 9 + c = -34 + c

So the equation becomes:

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

To write it in standard form, I need to move the constant term to the other side:

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

Now, comparing this to the standard equation $ (x - h)^2 + (y - k)^2 = r^2 $, I can see that the radius squared $ r^2 $ is equal to $ 34 - c $.

But the problem states that the radius should be 1. Therefore, $ r = 1 $, so $ r^2 = 1^2 = 1 $.

Setting the two expressions for $ r^2 $ equal:

$ 34 - c = 1 $

Solving for $ c $:

Subtract 34 from both sides:

$ -c = 1 - 34 $

$ -c = -33 $

Multiply both sides by -1:

$ c = 33 $

Wait, let me check my steps again to make sure I didn't make a mistake.

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

Completing the square:

For x: $ x^2 -10x $ becomes $ (x - 5)^2 -25 $

For y: $ y^2 +6y $ becomes $ (y + 3)^2 -9 $

So substituting back:

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

Combine constants: -25 -9 = -34, so:

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

Then moving -34 + c to the other side:

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

Yes, that's correct. Then since radius squared is 1, so 34 - c = 1, so c = 34 -1 = 33.

Wait, but let me think again. If the radius is 1, then the right-hand side must be 1. Therefore, 34 - c = 1, so c = 34 -1 = 33. That seems correct.

But let me check if I didn't mix up any signs. Let me re-express the equation again:

Original equation after completing the square:

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

So if the radius is 1, then 34 - c must equal 1. Therefore, solving for c:

c = 34 - 1 = 33.

Yes, that seems right.

But let me verify by plugging c = 33 back into the original equation and see if the radius is indeed 1.

Original equation with c = 33:

$ x^2 -10x + y^2 +6y +33 = 0 $

Completing the square again:

x terms: (x -5)^2 -25

y terms: (y +3)^2 -9

So equation becomes:

(x -5)^2 -25 + (y +3)^2 -9 +33 = 0

Calculating constants: -25 -9 +33 = (-34) +33 = -1

Therefore:

(x -5)^2 + (y +3)^2 -1 = 0 => (x -5)^2 + (y +3)^2 = 1

Which is a circle with center (5, -3) and radius 1. Perfect! That matches the requirement.

Therefore, c = 33 is correct.

Wait, but let me check if there's another way to approach this problem, just to confirm.

Alternatively, the general equation of a circle is $ x^2 + y^2 + 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, comparing to the given equation $ x^2 -10x + y^2 +6y + c = 0 $, we have D = -10, E = 6, F = c.

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

Calculating that:

$ (-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. Setting that equal to 1:

34 - c = 1 => c = 33.

Same result. So that confirms it again.

Therefore, I think the answer is 33.

But just to be thorough, let me check if I made any mistake in the sign when completing the square.

Original equation:

x² -10x + y² +6y +c = 0

Completing the square for x:

x² -10x. Take half of -10, which is -5, square it: 25. So add and subtract 25:

(x² -10x +25 -25) = (x -5)² -25

Similarly for y² +6y: half of 6 is 3, square is 9. So add and subtract 9:

(y² +6y +9 -9) = (y +3)² -9

Therefore, substituting back:

(x -5)² -25 + (y +3)² -9 + c = 0 => (x -5)² + (y +3)² -34 + c = 0 => (x -5)² + (y +3)² = 34 - c.

Yes, that's correct. So radius squared is 34 - c. Therefore, 34 - c = 1 => c = 33.

Another way to think about it: if the radius is 1, then the equation should be (x - h)^2 + (y - k)^2 = 1. Expanding that, we get x² - 2hx + h² + y² - 2ky + k² = 1. Which can be written as x² + y² - 2hx - 2ky + (h² + k² -1) = 0. Comparing this to the original equation x² -10x + y² +6y +c = 0, we can match coefficients.

So, coefficients of x: -2h = -10 => h = 5.

Coefficients of y: -2k = 6 => k = -3.

Then the constant term: h² + k² -1 = c.

Calculating h² + k² -1: 5² + (-3)^2 -1 = 25 + 9 -1 = 33. Therefore, c = 33.

Same answer again.

Therefore, regardless of the method, I get c = 33.

I think that's solid. So the answer is 33.

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

To determine the value of $ c $ such that the circle defined by the equation

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

has a radius of length 1, we begin by rewriting the equation in the standard form of a circle:

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

---

### Step 1: Complete the Square

We complete the square for both the $ x $ and $ y $ terms.

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

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

Substituting these 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 constant to the other side:

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

---

### Step 2: Compare to Standard Form

The standard form of a circle is:

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

From the equation above, we identify:

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

We are told the radius is 1, so:

$$
r^2 = 1^2 = 1
$$

Set the expressions for $ r^2 $ equal:

$$
34 - c = 1
$$

Solve for $ c $:

$$
c = 34 - 1 = 33
$$

---

### Final Answer

$$
\boxed{33}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of $ c $ such that the circle given by the equation $ x^2 - 10x + y^2 + 6y + c = 0 $ has a radius of length 1. Hmm, let me recall how to approach this.

First, I remember that the general equation of a circle is $ (x - h)^2 + (y - k)^2 = r^2 $, where $ (h, k) $ is the center and $ r $ is the radius. The given equation isn't in that standard form, so I probably need to complete the square for both the $ x $ and $ y $ terms. That should help me rewrite the equation into the standard form and then compare it to find the value of $ c $.

Let me start by rearranging the given equation. The original equation is:

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

I need to group the $ x $ terms and the $ y $ terms together. Let me rewrite it as:

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

Now, I need to complete the square for each of the quadratic terms. Let me handle the $ x $ terms first: $ x^2 - 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 can write:

$ x^2 - 10x = (x - 5)^2 - 25 $

Wait, let me check that. If I expand $ (x - 5)^2 $, it's $ x^2 - 10x + 25 $, so to get back to $ x^2 - 10x $, I need to subtract 25. Yes, that's correct. So, $ x^2 - 10x = (x - 5)^2 - 25 $.

Now, moving on to the $ y $ terms: $ y^2 + 6y $. Similarly, take the coefficient of $ y $, which is 6, divide by 2 to get 3, square it to get 9. Therefore:

$ y^2 + 6y = (y + 3)^2 - 9 $

Let me verify that. Expanding $ (y + 3)^2 $ gives $ y^2 + 6y + 9 $, so subtracting 9 gives back $ y^2 + 6y $. Correct.

So substituting these back into the original equation:

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

Now, let me combine the constants:

-25 - 9 + c = -34 + c

So the equation becomes:

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

To write it in standard form, I need to move the constant term to the other side:

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

Now, comparing this to the standard equation $ (x - h)^2 + (y - k)^2 = r^2 $, I can see that the radius squared $ r^2 $ is equal to $ 34 - c $.

But the problem states that the radius should be 1. Therefore, $ r = 1 $, so $ r^2 = 1^2 = 1 $.

Setting the two expressions for $ r^2 $ equal:

$ 34 - c = 1 $

Solving for $ c $:

Subtract 34 from both sides:

$ -c = 1 - 34 $

$ -c = -33 $

Multiply both sides by -1:

$ c = 33 $

Wait, let me check my steps again to make sure I didn't make a mistake.

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

Completing the square:

For x: $ x^2 -10x $ becomes $ (x - 5)^2 -25 $

For y: $ y^2 +6y $ becomes $ (y + 3)^2 -9 $

So substituting back:

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

Combine constants: -25 -9 = -34, so:

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

Then moving -34 + c to the other side:

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

Yes, that's correct. Then since radius squared is 1, so 34 - c = 1, so c = 34 -1 = 33.

Wait, but let me think again. If the radius is 1, then the right-hand side must be 1. Therefore, 34 - c = 1, so c = 34 -1 = 33. That seems correct.

But let me check if I didn't mix up any signs. Let me re-express the equation again:

Original equation after completing the square:

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

So if the radius is 1, then 34 - c must equal 1. Therefore, solving for c:

c = 34 - 1 = 33.

Yes, that seems right.

But let me verify by plugging c = 33 back into the original equation and see if the radius is indeed 1.

Original equation with c = 33:

$ x^2 -10x + y^2 +6y +33 = 0 $

Completing the square again:

x terms: (x -5)^2 -25

y terms: (y +3)^2 -9

So equation becomes:

(x -5)^2 -25 + (y +3)^2 -9 +33 = 0

Calculating constants: -25 -9 +33 = (-34) +33 = -1

Therefore:

(x -5)^2 + (y +3)^2 -1 = 0 => (x -5)^2 + (y +3)^2 = 1

Which is a circle with center (5, -3) and radius 1. Perfect! That matches the requirement.

Therefore, c = 33 is correct.

Wait, but let me check if there's another way to approach this problem, just to confirm.

Alternatively, the general equation of a circle is $ x^2 + y^2 + 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, comparing to the given equation $ x^2 -10x + y^2 +6y + c = 0 $, we have D = -10, E = 6, F = c.

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

Calculating that:

$ (-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. Setting that equal to 1:

34 - c = 1 => c = 33.

Same result. So that confirms it again.

Therefore, I think the answer is 33.

But just to be thorough, let me check if I made any mistake in the sign when completing the square.

Original equation:

x² -10x + y² +6y +c = 0

Completing the square for x:

x² -10x. Take half of -10, which is -5, square it: 25. So add and subtract 25:

(x² -10x +25 -25) = (x -5)² -25

Similarly for y² +6y: half of 6 is 3, square is 9. So add and subtract 9:

(y² +6y +9 -9) = (y +3)² -9

Therefore, substituting back:

(x -5)² -25 + (y +3)² -9 + c = 0 => (x -5)² + (y +3)² -34 + c = 0 => (x -5)² + (y +3)² = 34 - c.

Yes, that's correct. So radius squared is 34 - c. Therefore, 34 - c = 1 => c = 33.

Another way to think about it: if the radius is 1, then the equation should be (x - h)^2 + (y - k)^2 = 1. Expanding that, we get x² - 2hx + h² + y² - 2ky + k² = 1. Which can be written as x² + y² - 2hx - 2ky + (h² + k² -1) = 0. Comparing this to the original equation x² -10x + y² +6y +c = 0, we can match coefficients.

So, coefficients of x: -2h = -10 => h = 5.

Coefficients of y: -2k = 6 => k = -3.

Then the constant term: h² + k² -1 = c.

Calculating h² + k² -1: 5² + (-3)^2 -1 = 25 + 9 -1 = 33. Therefore, c = 33.

Same answer again.

Therefore, regardless of the method, I get c = 33.

I think that's solid. So the answer is 33.

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

To determine the value of $ c $ such that the circle defined by the equation

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

has a radius of length 1, we begin by rewriting the equation in the standard form of a circle:

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

---

### Step 1: Complete the Square

We complete the square for both the $ x $ and $ y $ terms.

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

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

Substituting these 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 constant to the other side:

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

---

### Step 2: Compare to Standard Form

The standard form of a circle is:

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

From the equation above, we identify:

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

We are told the radius is 1, so:

$$
r^2 = 1^2 = 1
$$

Set the expressions for $ r^2 $ equal:

$$
34 - c = 1
$$

Solve for $ c $:

$$
c = 34 - 1 = 33
$$

---

### 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 $c = \boxed{33}$.