💡Okay, so I have this problem about a triangular table where consecutive odd numbers are placed. The first row has one number, the second row has two numbers, the third row has three numbers, and so on. I need to prove that the sum of the numbers in the nth row is equal to n³. Hmm, interesting. Let me try to figure this out step by step.First, I should visualize what this triangular table looks like. The first row has just one number, which is 1. The second row has two numbers, which would be the next two odd numbers: 3 and 5. The third row has three numbers: 7, 9, and 11. The fourth row would then have 13, 15, 17, and 19, and so on. So, each row starts with the next odd number after the last number of the previous row.Now, I need to find the sum of the numbers in the nth row. Let's denote the nth row as Row n. The first thing I notice is that the numbers in each row form an arithmetic sequence because they are consecutive odd numbers. In an arithmetic sequence, each term increases by a constant difference, which in this case is 2.To find the sum of an arithmetic sequence, I remember the formula: S = n/2 * (first term + last term). So, if I can find the first term and the last term of the nth row, I can use this formula to find the sum.But before that, I need to figure out what the first term of the nth row is. Since the numbers are consecutive odd numbers, the first number in the nth row depends on how many numbers are there before the nth row. The total number of numbers before the nth row is the sum of the numbers of terms in the first (n-1) rows. That is, 1 + 2 + 3 + ... + (n-1). I recall that the sum of the first (n-1) natural numbers is given by (n-1)*n/2. So, the total number of terms before the nth row is (n-1)*n/2.Since each term is an odd number, the first term of the nth row would be the next odd number after the last term of the (n-1)th row. The last term of the (n-1)th row is the ((n-1)*n/2)th odd number. The nth odd number is given by 2n - 1, so the ((n-1)*n/2)th odd number is 2*((n-1)*n/2) + 1. Wait, let me think about that again.Actually, the kth odd number is 2k - 1. So, if there are (n-1)*n/2 numbers before the nth row, the last number of the (n-1)th row is 2*((n-1)*n/2) - 1. Simplifying that, it's (n-1)*n - 1. Therefore, the first term of the nth row is (n-1)*n - 1 + 2, which simplifies to n² - n + 1. Okay, so the first term of the nth row is n² - n + 1.Now, the last term of the nth row would be the first term plus 2*(n-1), since there are n terms in the row, each increasing by 2. So, the last term is (n² - n + 1) + 2*(n-1) = n² - n + 1 + 2n - 2 = n² + n - 1.Now that I have the first term (a = n² - n + 1) and the last term (l = n² + n - 1), I can use the arithmetic series sum formula. The sum S of the nth row is S = n/2 * (a + l). Plugging in the values, S = n/2 * (n² - n + 1 + n² + n - 1). Simplifying inside the parentheses: n² - n + 1 + n² + n - 1 = 2n². So, S = n/2 * 2n² = n³.Wait, that seems too straightforward. Let me double-check my steps. First, I calculated the total number of terms before the nth row as (n-1)*n/2. Then, the last term before the nth row is the ((n-1)*n/2)th odd number, which is 2*((n-1)*n/2) - 1 = (n-1)*n - 1. Therefore, the first term of the nth row is (n-1)*n - 1 + 2 = n² - n + 1. That seems correct.Then, the last term of the nth row is the first term plus 2*(n-1), which gives n² + n - 1. Adding the first and last terms: (n² - n + 1) + (n² + n - 1) = 2n². Multiplying by n/2 gives n³. Yeah, that checks out.Alternatively, maybe I can think of it another way to confirm. I know that the sum of the first m odd numbers is m². So, if I consider the sum up to the nth row, it would be the sum of the first (1 + 2 + 3 + ... + n) odd numbers, which is (n(n+1)/2)². But wait, that's the sum up to the nth row, not just the nth row itself.But if I subtract the sum up to the (n-1)th row from the sum up to the nth row, I should get the sum of the nth row. The sum up to the nth row is (n(n+1)/2)², and the sum up to the (n-1)th row is ((n-1)n/2)². So, the sum of the nth row is (n(n+1)/2)² - ((n-1)n/2)².Let me compute that: [n²(n+1)²/4] - [n²(n-1)²/4] = n²/4 [(n+1)² - (n-1)²]. Expanding the squares: (n² + 2n + 1) - (n² - 2n + 1) = 4n. So, the sum becomes n²/4 * 4n = n³. That's the same result as before. So, this alternative method also confirms that the sum of the nth row is n³.Hmm, so both methods lead to the same conclusion. That gives me more confidence that the sum of the numbers in the nth row is indeed n³.I think I've covered the necessary steps and even double-checked using a different approach. It seems solid. I don't see any mistakes in my reasoning, so I'm ready to present the proof.

💡Okay, so I have this determinant to compute, and it's a 3x3 matrix where each entry is a sine function of an integer. The matrix looks like this:[begin{vmatrix} sin 1 & sin 2 & sin 3 sin 4 & sin 5 & sin 6 sin 7 & sin 8 & sin 9 end{vmatrix}]All the angles are in radians, which is important because if they were in degrees, the values would be different. I remember that determinants can sometimes be zero if the columns (or rows) are linearly dependent. So, maybe I can check if the columns are linearly dependent here.Looking at the matrix, each row seems to follow a pattern. The first row has sin1, sin2, sin3. The second row has sin4, sin5, sin6, and the third row has sin7, sin8, sin9. So, each row is just the sine of consecutive integers starting from 1, 4, and 7 respectively.I wonder if there's a relationship between these sine functions that can help me. Maybe using some trigonometric identities? I recall that there are addition formulas for sine, like sin(a + b) = sin a cos b + cos a sin b. Maybe I can use that to express sin(n + 2) in terms of sin n and sin(n + 1).Let me try that. Let's take the first row: sin1, sin2, sin3. If I consider sin3, which is sin(2 + 1), I can write it as sin2 cos1 + cos2 sin1. Similarly, sin6 is sin(5 + 1) = sin5 cos1 + cos5 sin1, and sin9 is sin(8 + 1) = sin8 cos1 + cos8 sin1.So, each of the third column entries can be expressed as a combination of the previous two columns. That is, sin3 = sin2 cos1 + cos2 sin1, sin6 = sin5 cos1 + cos5 sin1, and sin9 = sin8 cos1 + cos8 sin1.Hmm, so the third column is a linear combination of the first and second columns. Specifically, each entry in the third column is equal to cos1 times the corresponding entry in the second column plus sin1 times the corresponding entry in the first column. Wait, is that right?Let me write it out:For the first row:sin3 = sin2 cos1 + cos2 sin1For the second row:sin6 = sin5 cos1 + cos5 sin1For the third row:sin9 = sin8 cos1 + cos8 sin1So, if I factor out cos1 and sin1, I can write the third column as:Third column = cos1 * second column + sin1 * first columnBut wait, that's not exactly accurate because the coefficients cos1 and sin1 are constants, not depending on the row. So, actually, the third column is a linear combination of the first and second columns with coefficients cos1 and sin1 respectively.Therefore, the third column is dependent on the first two columns. In linear algebra, if one column is a linear combination of the others, the determinant of the matrix is zero because the matrix is not full rank.So, does that mean the determinant is zero? Let me think again.Yes, because if the columns are linearly dependent, the determinant must be zero. I can confirm this by recalling that the determinant measures the volume of the parallelepiped spanned by the columns, and if they are linearly dependent, that volume is zero.Alternatively, I can compute the determinant using the rule of Sarrus or cofactor expansion, but that might be more time-consuming. Since I've established that the third column is a linear combination of the first two, it's sufficient to conclude that the determinant is zero.Just to be thorough, let me write out the relationship again:For each row i (where i = 1, 4, 7), the third element is sin(i + 2) = sin(i + 1) cos1 + cos(i + 1) sin1.So, in terms of the columns:Column3 = cos1 * Column2 + sin1 * Column1This is a clear linear dependence, so the determinant is zero.Therefore, the determinant of the given matrix is zero.

💡Okay, so I have this problem where I need to show that for a continuous closed curve ( C ) in the plane that doesn't cross itself, and a point ( Q ) inside ( C ), there exist two points ( P_1 ) and ( P_2 ) on ( C ) such that ( Q ) is the midpoint of the segment ( P_1P_2 ). Hmm, interesting. I remember something about midpoints and curves, maybe using some topological arguments or something like the Jordan curve theorem?First, let me try to visualize this. If ( C ) is a simple closed curve, it divides the plane into an interior and an exterior region. ( Q ) is inside, so it's in the bounded region enclosed by ( C ). I need to find two points on ( C ) such that their midpoint is ( Q ). That means if I draw a line segment between ( P_1 ) and ( P_2 ), it should pass through ( Q ) and ( Q ) should be exactly halfway between them.Maybe I can think about reflecting points across ( Q ). If I take a point ( P ) on ( C ), its reflection across ( Q ) would be another point ( P' ) such that ( Q ) is the midpoint of ( PP' ). So, if I can find a point ( P ) on ( C ) such that its reflection ( P' ) is also on ( C ), then I'm done. But how do I ensure that such a point exists?Wait, maybe I can use the idea of the curve being closed and continuous. Since ( C ) is a closed curve, it's compact, right? So, maybe I can use some fixed-point theorem or something like that. Or perhaps Brouwer's theorem? Hmm, not sure.Another thought: if I consider all possible points on ( C ), their reflections across ( Q ) would form another curve ( C' ). If ( C' ) intersects ( C ), then the intersection points would be the points ( P ) such that ( P' ) is also on ( C ). So, if I can show that ( C ) and ( C' ) must intersect, then I can find such points ( P_1 ) and ( P_2 ).But how do I show that ( C ) and ( C' ) intersect? Since ( C ) is a simple closed curve, ( C' ) is just another simple closed curve, but it's the reflection of ( C ) across ( Q ). Since ( Q ) is inside ( C ), reflecting ( C ) across ( Q ) would place ( C' ) such that it's still a closed curve, but maybe overlapping with ( C ) in some way.Wait, actually, reflecting ( C ) across ( Q ) would invert the orientation, but since ( C ) is closed, ( C' ) should also be a closed curve. Now, since ( Q ) is inside ( C ), reflecting ( C ) across ( Q ) would map the interior of ( C ) to the exterior of ( C' ) and vice versa. So, ( C ) and ( C' ) are two closed curves, one enclosing ( Q ) and the other being its reflection.But does this necessarily mean they intersect? I'm not sure. Maybe I need a different approach.Let me think about parametrizing the curve. Suppose I parametrize ( C ) as ( gamma(t) ) where ( t ) ranges from 0 to 1, and ( gamma(0) = gamma(1) ). Then, for each point ( gamma(t) ), its reflection across ( Q ) is ( 2Q - gamma(t) ). If I can show that the function ( f(t) = 2Q - gamma(t) ) intersects ( gamma(t) ) for some ( t ), then I have my points.So, define ( f(t) = 2Q - gamma(t) ). I need to find ( t ) such that ( f(t) = gamma(s) ) for some ( s ). That is, ( 2Q - gamma(t) = gamma(s) ). So, ( gamma(s) + gamma(t) = 2Q ). This is exactly the condition that ( Q ) is the midpoint of ( gamma(s) ) and ( gamma(t) ).But how do I ensure that such ( s ) and ( t ) exist? Maybe I can use the intermediate value theorem or something similar. If I can show that as ( t ) varies, ( f(t) ) must cross ( C ), then we're done.Alternatively, consider the function ( g(t) = gamma(t) + gamma(t + pi) ) or something like that, but I'm not sure if that applies here.Wait, another idea: consider the set of all midpoints between points on ( C ). Since ( C ) is compact, this set should also be compact. And since ( Q ) is inside ( C ), maybe this set covers ( Q ). But I'm not sure how to formalize that.Let me go back to the reflection idea. If I reflect ( C ) across ( Q ), getting ( C' ), then ( C ) and ( C' ) are both simple closed curves. Since ( Q ) is inside ( C ), ( C' ) must enclose the reflection of the interior of ( C ), which is the exterior of ( C ) with respect to ( Q ). So, ( C' ) is another curve that encloses a region, but is it possible that ( C ) and ( C' ) don't intersect?Hmm, if ( C ) is convex, then reflecting it across ( Q ) would result in ( C' ) also being convex, and they would intersect. But ( C ) is just a simple closed curve, not necessarily convex. So, maybe they don't intersect? But I think they must because of the way reflection works.Wait, actually, since ( Q ) is inside ( C ), reflecting ( C ) across ( Q ) would map the interior to the exterior, but the curve ( C ) itself is mapped to ( C' ). So, ( C ) and ( C' ) must intersect because otherwise, one would be entirely inside the other or entirely outside, but since ( C' ) is the reflection, it can't be entirely inside or outside without crossing ( C ).But I'm not entirely sure. Maybe I can use the Jordan curve theorem, which says that a simple closed curve divides the plane into two regions: interior and exterior. If ( C ) and ( C' ) are both simple closed curves, and ( C' ) is the reflection of ( C ), then ( C' ) must intersect ( C ) because otherwise, one would be entirely inside the other, which isn't possible since ( Q ) is inside ( C ) and ( C' ) is the reflection.Wait, actually, if ( C ) is the original curve, and ( C' ) is its reflection across ( Q ), then ( Q ) is inside ( C ), so the reflection ( C' ) would enclose the reflection of the interior of ( C ), which is the exterior of ( C ) with respect to ( Q ). So, ( C' ) must intersect ( C ) because otherwise, ( C' ) would be entirely inside or outside ( C ), but since it's the reflection, it can't be entirely inside or outside without crossing.Therefore, ( C ) and ( C' ) must intersect at least at two points. These intersection points would be points ( P ) such that ( P ) is on both ( C ) and ( C' ), meaning ( P ) is the reflection of itself across ( Q ), which would imply ( Q ) is the midpoint of ( P ) and ( P ), but that's trivial. Wait, no, actually, if ( P ) is on both ( C ) and ( C' ), then ( P ) is the reflection of some point ( P' ) on ( C ), so ( Q ) is the midpoint of ( P ) and ( P' ).Wait, that makes sense. So, if ( C ) and ( C' ) intersect at a point ( P ), then ( P ) is on ( C ), and its reflection ( P' ) is also on ( C ), so ( Q ) is the midpoint of ( P ) and ( P' ). Therefore, such points ( P_1 ) and ( P_2 ) exist.But I need to make sure that ( C ) and ( C' ) actually intersect. How can I formally show that?Maybe I can use the fact that ( C ) is a simple closed curve and ( C' ) is another simple closed curve, and they are related by reflection, which is a homeomorphism. Since ( Q ) is inside ( C ), reflecting ( C ) across ( Q ) would map the interior to the exterior, so ( C' ) must intersect ( C ) because otherwise, ( C' ) would be entirely inside or outside, which isn't possible due to the reflection.Alternatively, consider the function ( f(t) = gamma(t) + gamma(t) - 2Q ). Wait, no, that's not helpful. Maybe think about the vector from ( Q ) to a point on ( C ). For each point ( P ) on ( C ), define the vector ( overrightarrow{QP} ). Then, the reflection of ( P ) across ( Q ) is ( P' = 2Q - P ). So, if I can show that ( P' ) is also on ( C ), then ( Q ) is the midpoint.But how do I ensure that ( P' ) is on ( C )? Maybe by considering the function ( f(P) = 2Q - P ) and showing that it has a fixed point on ( C ). But fixed points would mean ( P = 2Q - P ), so ( 2P = 2Q ), meaning ( P = Q ), but ( Q ) is inside ( C ), not necessarily on ( C ).Hmm, maybe another approach. Let's consider the set ( S = { P in C mid 2Q - P in C } ). If I can show that ( S ) is non-empty, then we're done. But how?Alternatively, think about the function ( g: C to mathbb{R}^2 ) defined by ( g(P) = 2Q - P ). Since ( C ) is compact and ( g ) is continuous, the image ( g(C) ) is also compact. Now, ( g(C) ) is the reflection of ( C ) across ( Q ), which is ( C' ). So, ( C' ) is compact and closed.Now, since ( C ) and ( C' ) are both closed curves, and ( C' ) is the reflection of ( C ), they must intersect. If they didn't, then one would be entirely inside the other or entirely outside. But since ( Q ) is inside ( C ), reflecting ( C ) across ( Q ) would place ( C' ) such that it encloses the exterior of ( C ), so ( C' ) must intersect ( C ).Therefore, there exists a point ( P ) such that ( P in C ) and ( P in C' ). But ( P in C' ) means that ( P = 2Q - P' ) for some ( P' in C ). Therefore, ( Q ) is the midpoint of ( P ) and ( P' ).Wait, but does this guarantee that ( P ) and ( P' ) are distinct? If ( P = P' ), then ( Q ) would be the midpoint of ( P ) and itself, which is trivial, but we need two distinct points. So, I need to ensure that ( P neq P' ).If ( P = P' ), then ( P = 2Q - P ), so ( 2P = 2Q ), hence ( P = Q ). But ( Q ) is inside ( C ), not necessarily on ( C ). So, unless ( Q ) is on ( C ), which it isn't, ( P ) cannot equal ( P' ). Therefore, ( P ) and ( P' ) must be distinct points on ( C ).So, putting it all together, reflecting ( C ) across ( Q ) gives another curve ( C' ), and since ( C ) and ( C' ) must intersect, there exists a point ( P ) on both ( C ) and ( C' ). This means ( P ) is the reflection of some other point ( P' ) on ( C ), so ( Q ) is the midpoint of ( P ) and ( P' ).I think this makes sense. The key idea is using reflection and the properties of simple closed curves to ensure that the reflected curve intersects the original curve, giving the desired points.

💡Okay, so I have this problem about a geometric sequence. Let me try to figure it out step by step. First, the problem says that in the geometric sequence {a_n}, a_5 multiplied by a_13 equals 6. Also, a_4 plus a_14 equals 5. I need to find the value of a_80 divided by a_90. The options are given, so I can check my answer against them once I'm done.Alright, let's recall what a geometric sequence is. In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, usually denoted as q. So, the nth term of a geometric sequence can be written as a_n = a_1 * q^(n-1), where a_1 is the first term.Given that, let's write expressions for the terms mentioned in the problem.a_5 = a_1 * q^(5-1) = a_1 * q^4a_13 = a_1 * q^(13-1) = a_1 * q^12Similarly, a_4 = a_1 * q^3 and a_14 = a_1 * q^13.The problem states that a_5 * a_13 = 6. Let me compute that:a_5 * a_13 = (a_1 * q^4) * (a_1 * q^12) = a_1^2 * q^(4+12) = a_1^2 * q^16 = 6So, that's equation one: a_1^2 * q^16 = 6.Next, the problem says that a_4 + a_14 = 5. Let's write that:a_4 + a_14 = (a_1 * q^3) + (a_1 * q^13) = a_1 * q^3 + a_1 * q^13 = a_1 * q^3 (1 + q^10) = 5So, that's equation two: a_1 * q^3 (1 + q^10) = 5.Hmm, so I have two equations:1. a_1^2 * q^16 = 62. a_1 * q^3 (1 + q^10) = 5I need to solve for q or a_1. Maybe I can express a_1 from equation 1 and substitute into equation 2.From equation 1:a_1^2 = 6 / q^16So, a_1 = sqrt(6) / q^8But since we're dealing with real numbers, I can consider both positive and negative roots, but let's see if that's necessary.Substituting a_1 into equation 2:(sqrt(6) / q^8) * q^3 * (1 + q^10) = 5Simplify:sqrt(6) / q^5 * (1 + q^10) = 5So,sqrt(6) * (1 + q^10) / q^5 = 5Let me denote r = q^5. Then, q^10 = r^2.So, substituting:sqrt(6) * (1 + r^2) / r = 5Multiply both sides by r:sqrt(6) * (1 + r^2) = 5rBring all terms to one side:sqrt(6) * (1 + r^2) - 5r = 0So,sqrt(6) * r^2 - 5r + sqrt(6) = 0This is a quadratic equation in terms of r. Let's write it as:sqrt(6) * r^2 - 5r + sqrt(6) = 0Let me solve for r using the quadratic formula. For a quadratic equation ax^2 + bx + c = 0, the solutions are x = (-b ± sqrt(b^2 - 4ac)) / (2a).Here, a = sqrt(6), b = -5, c = sqrt(6).So,r = [5 ± sqrt(25 - 4 * sqrt(6) * sqrt(6))] / (2 * sqrt(6))Simplify the discriminant:sqrt(25 - 4 * sqrt(6) * sqrt(6)) = sqrt(25 - 4 * 6) = sqrt(25 - 24) = sqrt(1) = 1So,r = [5 ± 1] / (2 * sqrt(6))Thus, two solutions:r = (5 + 1) / (2 * sqrt(6)) = 6 / (2 * sqrt(6)) = 3 / sqrt(6) = sqrt(6)/2 after rationalizing.And,r = (5 - 1) / (2 * sqrt(6)) = 4 / (2 * sqrt(6)) = 2 / sqrt(6) = sqrt(6)/3 after rationalizing.So, r = sqrt(6)/2 or sqrt(6)/3.But remember, r = q^5. So,Case 1: q^5 = sqrt(6)/2Case 2: q^5 = sqrt(6)/3Now, I need to find a_80 / a_90.Let me express a_80 and a_90 in terms of a_1 and q.a_80 = a_1 * q^(79)a_90 = a_1 * q^(89)So,a_80 / a_90 = (a_1 * q^79) / (a_1 * q^89) = q^(79 - 89) = q^(-10) = 1 / q^10So, I need to find 1 / q^10.But from earlier, we have r = q^5, so q^10 = (q^5)^2 = r^2.Therefore,a_80 / a_90 = 1 / r^2So, in each case:Case 1: r = sqrt(6)/2, so r^2 = (6)/4 = 3/2. Therefore, 1 / r^2 = 2/3.Case 2: r = sqrt(6)/3, so r^2 = (6)/9 = 2/3. Therefore, 1 / r^2 = 3/2.So, the ratio a_80 / a_90 is either 2/3 or 3/2.Looking back at the options:A: 2/3 or 3/2B: 3 or -2C: 2/3D: 3/2So, the correct answer is A.Wait, but let me double-check my steps to ensure I didn't make a mistake.Starting from the beginning:Given a_5 * a_13 = 6, which I correctly expanded as a_1^2 * q^16 = 6.Then, a_4 + a_14 = 5, which I wrote as a_1 * q^3 (1 + q^10) = 5.Expressed a_1 from the first equation as sqrt(6)/q^8, substituted into the second equation, leading to sqrt(6)*(1 + q^10)/q^5 = 5.Set r = q^5, so equation becomes sqrt(6)*(1 + r^2)/r = 5, which simplifies to sqrt(6)*(1 + r^2) = 5r.Then, quadratic equation sqrt(6)*r^2 -5r + sqrt(6) = 0.Solutions via quadratic formula: [5 ± sqrt(25 - 24)] / (2*sqrt(6)) = [5 ±1]/(2*sqrt(6)).Thus, r = 6/(2*sqrt(6)) = sqrt(6)/2 and r = 4/(2*sqrt(6)) = sqrt(6)/3.Therefore, q^5 = sqrt(6)/2 or sqrt(6)/3.Then, a_80 / a_90 = 1/q^10 = 1/(q^5)^2 = 1/( (sqrt(6)/2)^2 ) = 1/(6/4) = 4/6 = 2/3.Similarly, 1/( (sqrt(6)/3)^2 ) = 1/(6/9) = 9/6 = 3/2.So, yes, the ratio is either 2/3 or 3/2, which is option A.I think that's correct. So, the answer should be A.**Final Answer**boxed{A}

💡Okay, so I need to list all the divisors of 30 and 135 and then count how many there are for each number. Hmm, divisors are numbers that divide another number completely without leaving any remainder. Let me start with 30.First, I'll try to list the divisors of 30. I know that 1 and 30 are always divisors because any number is divisible by 1 and itself. Then, I'll check the numbers between 1 and 30 to see which ones divide 30 evenly.Let's see:- 2 divides 30 because 30 ÷ 2 = 15, which is an integer.- 3 divides 30 because 30 ÷ 3 = 10, which is an integer.- 5 divides 30 because 30 ÷ 5 = 6, which is an integer.- 6 divides 30 because 30 ÷ 6 = 5, which is an integer.- 10 divides 30 because 30 ÷ 10 = 3, which is an integer.- 15 divides 30 because 30 ÷ 15 = 2, which is an integer.So, the divisors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30. That makes 8 divisors in total.Now, moving on to 135. I'll use a similar approach. Start with 1 and 135, then check the numbers in between.- 3 divides 135 because 135 ÷ 3 = 45, which is an integer.- 5 divides 135 because 135 ÷ 5 = 27, which is an integer.- 9 divides 135 because 135 ÷ 9 = 15, which is an integer.- 15 divides 135 because 135 ÷ 15 = 9, which is an integer.- 27 divides 135 because 135 ÷ 27 = 5, which is an integer.- 45 divides 135 because 135 ÷ 45 = 3, which is an integer.So, the divisors of 135 are: 1, 3, 5, 9, 15, 27, 45, and 135. That also makes 8 divisors in total.Wait a minute, both 30 and 135 have 8 divisors each. Is that just a coincidence, or is there something more to it? Maybe it has to do with their prime factorizations.Let me think about the prime factors of 30. Breaking it down:- 30 can be divided by 2: 30 ÷ 2 = 15- 15 can be divided by 3: 15 ÷ 3 = 5- 5 is a prime number.So, the prime factors of 30 are 2, 3, and 5. Each of these primes appears once in the factorization. There's a formula for finding the number of divisors based on prime factors. If a number N can be expressed as ( N = p_1^{a} times p_2^{b} times ldots times p_k^{c} ), then the number of divisors is ( (a+1)(b+1)ldots(c+1) ).Applying this to 30:- ( 30 = 2^1 times 3^1 times 5^1 )- Number of divisors = ( (1+1)(1+1)(1+1) = 2 times 2 times 2 = 8 )That matches what I found earlier.Now, for 135:- 135 can be divided by 3: 135 ÷ 3 = 45- 45 can be divided by 3: 45 ÷ 3 = 15- 15 can be divided by 3: 15 ÷ 3 = 5- 5 is a prime number.So, the prime factors of 135 are 3 and 5, with 3 appearing three times and 5 appearing once. Using the same formula:- ( 135 = 3^3 times 5^1 )- Number of divisors = ( (3+1)(1+1) = 4 times 2 = 8 )Again, that matches the count I got by listing them out.It's interesting that both numbers have the same number of divisors even though their prime factors are different. I guess it's because the combination of exponents in their prime factorizations leads to the same product when applying the formula.I wonder if there's a way to predict this without listing all the divisors. Maybe by just looking at the exponents in the prime factorization. For example, if two numbers have prime factors raised to exponents that, when each is increased by one and multiplied together, give the same result, they will have the same number of divisors.In this case:- For 30: exponents are 1, 1, 1. So, ( (1+1)(1+1)(1+1) = 8 )- For 135: exponents are 3 and 1. So, ( (3+1)(1+1) = 8 )Both give 8, hence both have 8 divisors.This makes me think about how the number of divisors is determined by the exponents in the prime factorization rather than the actual primes themselves. So, different numbers can have the same number of divisors if their prime exponents, when adjusted by adding one and multiplying, result in the same product.I should also consider if there's a connection between the two numbers, like their greatest common divisor (GCD) or least common multiple (LCM), but that might be a bit beyond what I need to do right now. For this problem, listing the divisors and counting them seems sufficient.Just to double-check, I'll recount the divisors for both numbers to make sure I didn't miss any or count any extra.For 30:1, 2, 3, 5, 6, 10, 15, 30. That's 8 numbers.For 135:1, 3, 5, 9, 15, 27, 45, 135. That's also 8 numbers.Okay, I'm confident that both have 8 divisors each.I think I've covered everything the problem asked for: listing the divisors and counting them. It was a good exercise to see how prime factorization relates to the number of divisors. It also shows how different numbers can have the same number of divisors based on their prime exponents.

💡Okay, so I need to figure out how much money the Razorback shop made from selling each jersey. Let's see, they sold 74 t-shirts and 156 jerseys during the game. They made a total of 25,740 from selling the jerseys. Hmm, I think I need to find out how much they made per jersey. First, I know that they sold 156 jerseys and made 25,740 from them. So, if I want to find out how much they made per jersey, I should divide the total amount made from jerseys by the number of jerseys sold. That makes sense because if you have a total amount and you want to find out the amount per item, you divide the total by the number of items.So, the formula I think I need is:Amount made per jersey = Total revenue from jerseys / Number of jerseys soldPlugging in the numbers:Amount made per jersey = 25,740 / 156Now, I need to do this division. Let me see, 25,740 divided by 156. Hmm, that might be a bit tricky. Maybe I can simplify this division by breaking it down.First, I can see that both numbers are divisible by 12. Let me check:25,740 ÷ 12 = 2,145156 ÷ 12 = 13So now, the problem becomes 2,145 divided by 13. That seems easier.2,145 ÷ 13Let me do this step by step. 13 times 100 is 1,300. Subtracting that from 2,145 gives me 845.Now, 13 times 60 is 780. Subtracting that from 845 gives me 65.13 times 5 is 65. So, adding up the multiples: 100 + 60 + 5 = 165.So, 2,145 ÷ 13 = 165.Therefore, 25,740 ÷ 156 = 165.Wait, let me double-check that to make sure I didn't make a mistake. If I multiply 156 by 165, I should get back to 25,740.156 × 165Let me break this down:156 × 100 = 15,600156 × 60 = 9,360156 × 5 = 780Now, adding those together: 15,600 + 9,360 = 24,96024,960 + 780 = 25,740Yes, that checks out. So, the amount made per jersey is 165.But wait, the problem also mentions that they make 67 off each t-shirt. Do I need to consider that in this calculation? Hmm, the question specifically asks how much they made from selling each jersey, and they already provided the total revenue from jerseys. So, I think I don't need to involve the t-shirt information here. It might be extra information or relevant for a different part of the problem, but for this particular question, I can focus solely on the jersey sales.Just to be thorough, if I wanted to find the total revenue from both t-shirts and jerseys, I would calculate:Total revenue from t-shirts = Number of t-shirts sold × Revenue per t-shirtTotal revenue from t-shirts = 74 × 67Let me calculate that:74 × 60 = 4,44074 × 7 = 518Adding those together: 4,440 + 518 = 4,958So, total revenue from t-shirts is 4,958.Adding that to the total revenue from jerseys: 4,958 + 25,740 = 30,698But since the question only asks about the revenue per jersey, I don't need this additional information. It's just extra practice to ensure I understand how to calculate total revenue.In summary, to find out how much they made from each jersey, I divided the total revenue from jerseys by the number of jerseys sold, which gave me 165 per jersey. I double-checked my calculations by multiplying 156 by 165 to ensure it equals 25,740, and it did. So, I'm confident that the amount made from each jersey is 165.