💡Okay, so I have this problem where I want to plant 45 sunflower plants, 81 corn plants, and 63 tomato plants in my garden. The goal is to arrange them in rows such that each row has the same number of plants, and each row contains only one type of plant. I need to figure out the greatest number of plants I can put in one row. Hmm, let me think about this. It seems like I need to find a common number that can divide all three quantities evenly. That way, I can have the same number of plants in each row without mixing different types of plants. So, maybe I should look into something called the greatest common divisor (GCD). I remember that the GCD is the largest number that can divide multiple numbers without leaving a remainder. That sounds like exactly what I need here.Alright, so to find the GCD of 45, 81, and 63, I should probably start by breaking each number down into its prime factors. Prime factorization might help me identify the common factors among these numbers. Let me try that.Starting with 45: I know that 45 divided by 5 is 9, and 9 is 3 times 3. So, the prime factors of 45 are 3, 3, and 5. In other words, 45 = 3^2 * 5.Next, 81: I recall that 81 is 9 times 9, and 9 is 3 times 3. So, breaking it down, 81 = 3^4.Now, 63: Let's see, 63 divided by 7 is 9, and 9 is 3 times 3. So, the prime factors of 63 are 3, 3, and 7. Therefore, 63 = 3^2 * 7.Okay, so now I have the prime factors:- 45 = 3^2 * 5- 81 = 3^4- 63 = 3^2 * 7Looking at these, the common prime factor among all three numbers is 3. Now, I need to find the smallest power of 3 that is present in all three factorizations. For 45, it's 3^2.For 81, it's 3^4.For 63, it's 3^2.The smallest power here is 3^2, which is 9. So, the greatest common divisor of 45, 81, and 63 is 9.This means that I can arrange the plants in rows of 9. Let me verify this to make sure it makes sense.For the sunflowers: 45 divided by 9 is 5. So, I can have 5 rows of sunflowers with 9 plants each.For the corn: 81 divided by 9 is 9. So, I can have 9 rows of corn with 9 plants each.For the tomatoes: 63 divided by 9 is 7. So, I can have 7 rows of tomatoes with 9 plants each.That seems to work out perfectly. Each row has the same number of plants, and each row contains only one type of plant. There are no leftover plants, which is ideal.I think I've got it right. The greatest number of plants I can put in one row is 9.

💡First, I need to determine the total weight of the blue weights. Each blue weight is 2 pounds, and Harry used 4 of them. So, multiplying 2 pounds by 4 gives me 8 pounds for the blue weights.Next, I'll calculate the total weight of the green weights. Each green weight is 3 pounds, and Harry used 5 of them. Multiplying 3 pounds by 5 results in 15 pounds for the green weights.Additionally, the metal bar itself weighs 2 pounds.Finally, I'll add up all these weights: 8 pounds from the blue weights, 15 pounds from the green weights, and 2 pounds from the metal bar. Adding these together gives a total weight of 25 pounds.

💡Okay, so I need to figure out for which positive integer values of ( n ) the expression ( 2^n + 1 ) is divisible by 3. Hmm, let me think about how to approach this.First, I remember that when dealing with divisibility by 3, it's often helpful to look at things modulo 3. So maybe I can analyze ( 2^n + 1 ) modulo 3 and see when it equals zero. That would mean ( 2^n + 1 equiv 0 pmod{3} ), right?Let me start by computing ( 2^n ) modulo 3 for small values of ( n ) to see if there's a pattern.- When ( n = 1 ): ( 2^1 = 2 ), so ( 2 + 1 = 3 ), which is divisible by 3. So ( n = 1 ) works.- When ( n = 2 ): ( 2^2 = 4 ), so ( 4 + 1 = 5 ). 5 divided by 3 leaves a remainder of 2, so ( n = 2 ) doesn't work.- When ( n = 3 ): ( 2^3 = 8 ), so ( 8 + 1 = 9 ), which is divisible by 3. So ( n = 3 ) works.- When ( n = 4 ): ( 2^4 = 16 ), so ( 16 + 1 = 17 ). 17 divided by 3 leaves a remainder of 2, so ( n = 4 ) doesn't work.- When ( n = 5 ): ( 2^5 = 32 ), so ( 32 + 1 = 33 ), which is divisible by 3. So ( n = 5 ) works.Hmm, I see a pattern here. It seems like when ( n ) is odd, ( 2^n + 1 ) is divisible by 3, and when ( n ) is even, it isn't. Let me check a few more to be sure.- When ( n = 6 ): ( 2^6 = 64 ), so ( 64 + 1 = 65 ). 65 divided by 3 leaves a remainder of 2, so ( n = 6 ) doesn't work.- When ( n = 7 ): ( 2^7 = 128 ), so ( 128 + 1 = 129 ), which is divisible by 3. So ( n = 7 ) works.Okay, the pattern seems consistent. Now, I should try to understand why this happens. Maybe by looking at the powers of 2 modulo 3.Let's compute ( 2^n mod 3 ):- ( 2^1 mod 3 = 2 )- ( 2^2 mod 3 = 4 mod 3 = 1 )- ( 2^3 mod 3 = 8 mod 3 = 2 )- ( 2^4 mod 3 = 16 mod 3 = 1 )- ( 2^5 mod 3 = 32 mod 3 = 2 )- ( 2^6 mod 3 = 64 mod 3 = 1 )So, it looks like ( 2^n mod 3 ) cycles between 2 and 1 depending on whether ( n ) is odd or even. Specifically, when ( n ) is odd, ( 2^n equiv 2 pmod{3} ), and when ( n ) is even, ( 2^n equiv 1 pmod{3} ).Therefore, ( 2^n + 1 mod 3 ) will be:- If ( n ) is odd: ( 2 + 1 = 3 equiv 0 pmod{3} )- If ( n ) is even: ( 1 + 1 = 2 equiv 2 pmod{3} )So, ( 2^n + 1 ) is divisible by 3 exactly when ( n ) is odd.Wait, but let me make sure I'm not missing something. Is there a mathematical way to generalize this beyond just checking small cases?I recall that for any integer ( a ) and modulus ( m ), ( a^n mod m ) can often be analyzed using Euler's theorem or Fermat's little theorem, but since 2 and 3 are coprime, maybe Euler's theorem applies here.Euler's theorem states that if ( a ) and ( m ) are coprime, then ( a^{phi(m)} equiv 1 pmod{m} ), where ( phi ) is Euler's totient function. For ( m = 3 ), ( phi(3) = 2 ), so ( 2^2 equiv 1 pmod{3} ).This means that the powers of 2 modulo 3 repeat every 2 steps. So, the cycle length is 2. Therefore, ( 2^n mod 3 ) cycles between 2 and 1 as ( n ) increases.Thus, for any odd ( n ), ( 2^n equiv 2 pmod{3} ), and adding 1 gives ( 2 + 1 = 3 equiv 0 pmod{3} ). For any even ( n ), ( 2^n equiv 1 pmod{3} ), and adding 1 gives ( 1 + 1 = 2 pmod{3} ), which isn't divisible by 3.So, this confirms the pattern I noticed earlier. Therefore, the expression ( 2^n + 1 ) is divisible by 3 if and only if ( n ) is an odd positive integer.I think that's a solid conclusion. I don't see any flaws in this reasoning, and the pattern holds for the cases I checked. So, I can confidently say that all positive integer values of ( n ) for which ( 2^n + 1 ) is divisible by 3 are the odd integers.

💡Alright, so I have this trigonometry problem here, and I need to find (cosleft(beta + frac{pi}{4}right)) given some other information. Let me try to unpack this step by step.First, let's list out what's given:1. Both (alpha) and (beta) are in the interval (left(frac{3pi}{4}, piright)). That means each of them is in the second quadrant, right? Because (frac{3pi}{4}) is 135 degrees, and (pi) is 180 degrees. So both angles are between 135 and 180 degrees.2. (cos(alpha + beta) = frac{4}{5}). Hmm, okay. So the cosine of the sum of (alpha) and (beta) is positive. Now, since (alpha) and (beta) are both in the second quadrant, their sum (alpha + beta) would be in the third or fourth quadrant, depending on how big they are. Let me think: if each is between 135 and 180 degrees, their sum would be between 270 and 360 degrees, right? So (alpha + beta) is in the fourth quadrant because 270 to 360 is the fourth quadrant. In the fourth quadrant, cosine is positive and sine is negative. That's good to know.3. (sinleft(alpha - frac{pi}{4}right) = frac{12}{13}). So the sine of (alpha) minus 45 degrees (since (frac{pi}{4}) is 45 degrees) is positive. Let me figure out where (alpha - frac{pi}{4}) is. Since (alpha) is between 135 and 180 degrees, subtracting 45 degrees would put it between 90 and 135 degrees. So that's the second quadrant. In the second quadrant, sine is positive, which matches the given value. So that makes sense.Okay, so I need to find (cosleft(beta + frac{pi}{4}right)). Let me think about how to approach this. Maybe I can use some trigonometric identities to express this in terms of the given information.I know that (cos(A + B) = cos A cos B - sin A sin B). So maybe if I can express (beta + frac{pi}{4}) in terms of (alpha) and (beta), I can use that identity.Wait, but I don't have information about (beta) directly. I have information about (alpha + beta) and (alpha - frac{pi}{4}). Maybe I can find (alpha) and (beta) separately and then compute (cosleft(beta + frac{pi}{4}right)). But that might be complicated. Alternatively, maybe I can express (beta + frac{pi}{4}) in terms of (alpha + beta) and (alpha - frac{pi}{4}).Let me try that. Let me denote:Let (A = alpha + beta) and (B = alpha - frac{pi}{4}). Then, I need to find (cosleft(beta + frac{pi}{4}right)). Let me see if I can express (beta + frac{pi}{4}) in terms of A and B.From (A = alpha + beta), we can write (beta = A - alpha). Then, (beta + frac{pi}{4} = A - alpha + frac{pi}{4}). Hmm, not sure if that's helpful.Alternatively, maybe I can solve for (alpha) and (beta) individually. Let me see if that's possible.From (A = alpha + beta), and (B = alpha - frac{pi}{4}), maybe I can solve for (alpha) and (beta). Let me try.From (B = alpha - frac{pi}{4}), we can write (alpha = B + frac{pi}{4}). Then, substituting into (A = alpha + beta), we get:(A = (B + frac{pi}{4}) + beta)So, (A = B + frac{pi}{4} + beta)Therefore, (beta = A - B - frac{pi}{4})So, (beta + frac{pi}{4} = A - B - frac{pi}{4} + frac{pi}{4} = A - B)So, (cosleft(beta + frac{pi}{4}right) = cos(A - B))That's interesting. So, (cosleft(beta + frac{pi}{4}right) = cos(A - B)), where (A = alpha + beta) and (B = alpha - frac{pi}{4}).So, I can use the cosine of difference identity:(cos(A - B) = cos A cos B + sin A sin B)So, I need to find (cos A), (cos B), (sin A), and (sin B).Given:- (A = alpha + beta), and we know (cos A = frac{4}{5}). So, we can find (sin A) using the Pythagorean identity.- (B = alpha - frac{pi}{4}), and we know (sin B = frac{12}{13}). So, we can find (cos B) using the Pythagorean identity.Let's compute (sin A) first.Since (cos A = frac{4}{5}), we have:(sin^2 A + cos^2 A = 1)So, (sin^2 A = 1 - cos^2 A = 1 - left(frac{4}{5}right)^2 = 1 - frac{16}{25} = frac{9}{25})Therefore, (sin A = pm frac{3}{5})But since (A = alpha + beta) is in the fourth quadrant (as we established earlier), sine is negative there. So, (sin A = -frac{3}{5})Next, for (B = alpha - frac{pi}{4}), we know (sin B = frac{12}{13}). Let's find (cos B).Again, using the Pythagorean identity:(sin^2 B + cos^2 B = 1)So, (cos^2 B = 1 - sin^2 B = 1 - left(frac{12}{13}right)^2 = 1 - frac{144}{169} = frac{25}{169})Therefore, (cos B = pm frac{5}{13})Now, we need to determine the sign of (cos B). Since (B = alpha - frac{pi}{4}) is in the second quadrant (as we saw earlier), cosine is negative there. So, (cos B = -frac{5}{13})Now, we have all the components needed to compute (cos(A - B)):(cos(A - B) = cos A cos B + sin A sin B)Plugging in the known values:(cos(A - B) = left(frac{4}{5}right)left(-frac{5}{13}right) + left(-frac{3}{5}right)left(frac{12}{13}right))Let me compute each term separately:First term: (frac{4}{5} times -frac{5}{13} = -frac{20}{65})Second term: (-frac{3}{5} times frac{12}{13} = -frac{36}{65})Adding them together:(-frac{20}{65} - frac{36}{65} = -frac{56}{65})So, (cosleft(beta + frac{pi}{4}right) = -frac{56}{65})Wait, let me double-check my calculations to make sure I didn't make a mistake.First, (cos A = frac{4}{5}), correct. (sin A = -frac{3}{5}), correct because A is in the fourth quadrant.For B, (sin B = frac{12}{13}), so (cos B = -frac{5}{13}), correct because B is in the second quadrant.Then, (cos(A - B) = cos A cos B + sin A sin B):(frac{4}{5} times -frac{5}{13} = -frac{20}{65})(-frac{3}{5} times frac{12}{13} = -frac{36}{65})Adding them: (-frac{20}{65} - frac{36}{65} = -frac{56}{65})Yes, that seems correct.So, the final answer is (-frac{56}{65}).**Final Answer**boxed{-dfrac{56}{65}}

💡First, I need to evaluate the given expression step by step to find the unknown number. The expression is:(0.86)^3 - (0.1)^3 / (0.86)^2 + some number + (0.1)^2 = 0.76I'll start by calculating the cubes and squares of the numbers involved.Calculating the cubes:- (0.86)^3 = 0.86 * 0.86 * 0.86 = 0.636056- (0.1)^3 = 0.1 * 0.1 * 0.1 = 0.001Calculating the squares:- (0.86)^2 = 0.86 * 0.86 = 0.7396- (0.1)^2 = 0.1 * 0.1 = 0.01Now, substitute these values back into the expression:0.636056 - 0.001 / 0.7396 + some number + 0.01 = 0.76Next, perform the division:0.001 / 0.7396 ≈ 0.001352Now, the expression becomes:0.636056 - 0.001352 + some number + 0.01 = 0.76Combine the constants:0.636056 - 0.001352 + 0.01 ≈ 0.644704So, the equation simplifies to:0.644704 + some number = 0.76To find the unknown number, subtract 0.644704 from 0.76:some number = 0.76 - 0.644704 ≈ 0.115296Therefore, the unknown number is approximately 0.115296.

💡First, let's understand the problem. The gauge indicates that there are 75 cups of liquid left when the tank is 50% full. This means that 75 cups represent half of the tank's total capacity.To find the total capacity, we can set up the equation:0.50 × Total Capacity = 75Solving for the Total Capacity:Total Capacity = 75 / 0.50Total Capacity = 150Therefore, the tank holds 150 cups when it is completely full.