answer：Okay, so I have this logic puzzle with four defendants: A, B, C, and D. There are four statements given, and I need to figure out who is definitely guilty and who is still in doubt. Let me try to break this down step by step.First, let me list out the given statements to have them clear in my mind:1) If A and B are both guilty, then C was an accomplice.2) If A is guilty, then at least one of the accused B or C was an accomplice.3) If C is guilty, then D was an accomplice.4) If A is not guilty, then D is guilty.Hmm, so these are all conditional statements. I think I need to analyze each statement and see how they relate to each other. Maybe I can use some logical deductions here.Let me start by considering the possibilities for A's guilt because statements 2 and 4 both involve A. If I can figure out whether A is guilty or not, that might help me with the others.**Case 1: Suppose A is guilty.**If A is guilty, then according to statement 2, at least one of B or C is an accomplice. So, either B is guilty, or C is guilty, or both.Now, let's see what else we know. If A is guilty, does that affect the other statements? Statement 1 says that if both A and B are guilty, then C was an accomplice. So, if A is guilty and B is also guilty, then C must be guilty. But if A is guilty and B is not guilty, then statement 2 still holds because C would have to be guilty.Wait, so if A is guilty, regardless of B's guilt, C might be guilty? Let me think.If A is guilty, then from statement 2, either B or C is guilty. If B is guilty, then from statement 1, C must be guilty. If B is not guilty, then C must be guilty to satisfy statement 2. So, in either case, if A is guilty, then C is guilty.So, if A is guilty, then C is guilty. Then, from statement 3, if C is guilty, then D is guilty. So, if A is guilty, then D is guilty as well.So, in this case, if A is guilty, then C and D are definitely guilty. What about B? If A is guilty, from statement 2, at least one of B or C is guilty. But since we already concluded that C is guilty, B could be either guilty or not guilty. So, B's guilt is in doubt in this case.So, under Case 1, A is guilty, C is guilty, D is guilty, and B is in doubt.**Case 2: Suppose A is not guilty.**If A is not guilty, then from statement 4, D is guilty. So, D is definitely guilty.Now, what about the others? Since A is not guilty, statements 1, 2, and 3 don't directly involve A anymore. Let's see:Statement 1: If A and B are both guilty, then C was an accomplice. But since A is not guilty, the condition is false, so this statement doesn't give us any information.Statement 2: If A is guilty, then at least one of B or C is guilty. Since A is not guilty, this statement is also irrelevant.Statement 3: If C is guilty, then D is guilty. We already know D is guilty, so if C is guilty, that's consistent. But if C is not guilty, that's also fine because the statement is only about if C is guilty.So, in this case, A is not guilty, D is guilty. What about B and C? They could be guilty or not guilty. There's no information that forces their guilt or innocence. So, both B and C are in doubt.Wait, but let me double-check. If A is not guilty, is there any way B or C's guilt can be determined? Statement 3 says if C is guilty, then D is guilty. Since D is guilty, C could be guilty or not. There's no contradiction if C is innocent because the implication only goes one way.Similarly, statement 1 and 2 don't impose any constraints because they start with A being guilty, which isn't the case here. So, yeah, B and C are in doubt.So, under Case 2, A is not guilty, D is guilty, and B and C are in doubt.Now, let's summarize both cases:- If A is guilty: A, C, D are guilty; B is in doubt.- If A is not guilty: D is guilty; A, B, C are in doubt.But wait, the question is asking who is definitely guilty and who is in doubt. So, regardless of whether A is guilty or not, D is guilty in both cases. So, D is definitely guilty.What about A? In Case 1, A is guilty; in Case 2, A is not guilty. So, A's guilt is in doubt.Similarly, C: In Case 1, C is guilty; in Case 2, C could be guilty or not. So, C is in doubt.B: In Case 1, B is in doubt; in Case 2, B is in doubt as well. So, B is definitely in doubt.Wait, hold on. Let me think again. In Case 1, B could be guilty or not, but in Case 2, B could be guilty or not. So, B is in doubt in both scenarios. So, B is definitely in doubt.But wait, C is in doubt in both scenarios as well? Wait, no. In Case 1, C is definitely guilty; in Case 2, C could be guilty or not. So, C is guilty in one case, and in doubt in another. So, overall, C is not definitely guilty, but could be guilty or not.Similarly, A is guilty in one case, not guilty in another.So, the only one who is definitely guilty is D.Wait, but hold on. Let me check if there's any other possibilities.Is there a scenario where D is not guilty? Let's see.If D is not guilty, then from statement 4, if A is not guilty, then D is guilty. So, if D is not guilty, then A must be guilty.So, if D is not guilty, A is guilty.But if A is guilty, then from statement 2, at least one of B or C is guilty.If A is guilty, and D is not guilty, let's see:From statement 3: If C is guilty, then D is guilty. But D is not guilty, so C cannot be guilty. So, C is not guilty.From statement 2: If A is guilty, then at least one of B or C is guilty. Since C is not guilty, B must be guilty.From statement 1: If A and B are both guilty, then C was an accomplice. But we just concluded that C is not guilty. So, this would be a contradiction because if A and B are guilty, then C must be guilty, but we have C not guilty.Therefore, this scenario is impossible. So, D cannot be not guilty. Therefore, D must be guilty.So, D is definitely guilty.What about A? If D is guilty, does that affect A? From statement 4: If A is not guilty, then D is guilty. So, D being guilty doesn't necessarily mean A is guilty or not. It just says that if A is not guilty, then D is guilty. But D can be guilty regardless of A's guilt.Wait, but earlier, when we supposed A is guilty, D is guilty. When we supposed A is not guilty, D is guilty. So, D is guilty in both cases.But when we supposed A is guilty, we had C guilty as well. When A is not guilty, C could be guilty or not.But wait, in the case where A is guilty, C is guilty, so D is guilty. In the case where A is not guilty, D is guilty regardless of C.So, D is definitely guilty.What about C? Is there a way to determine C's guilt?If A is guilty, C is guilty. If A is not guilty, C could be guilty or not. So, C is guilty in one scenario, in doubt in another. So, C is not definitely guilty.Similarly, B is in doubt in both scenarios.A is guilty in one scenario, not guilty in another.So, only D is definitely guilty.Wait, but hold on. Let me think if there's a way that C could be definitely guilty.If A is guilty, C is guilty. If A is not guilty, C could be guilty or not. So, C is guilty only if A is guilty. But since A could be guilty or not, C is not necessarily guilty.Similarly, B is guilty only if A is guilty and B is guilty, but since B's guilt is in doubt even when A is guilty, B is in doubt.So, yeah, only D is definitely guilty.Wait, but let me check another angle. Suppose A is guilty, then C is guilty, which makes D guilty. If A is not guilty, D is guilty regardless. So, D is definitely guilty.But what about C? If A is guilty, C is guilty. If A is not guilty, C could be guilty or not. So, C is guilty only if A is guilty. So, C's guilt is conditional on A's guilt.But since A's guilt is in doubt, C's guilt is also in doubt.Similarly, B is only guilty if A is guilty and B is guilty, but even if A is guilty, B might not be. So, B is in doubt.So, yeah, only D is definitely guilty.Wait, but let me think if there's any other constraints I might have missed.Statement 3: If C is guilty, then D is guilty. Since D is guilty, C could be guilty or not. So, C could be guilty or innocent, but D is guilty regardless.So, yeah, C is in doubt.So, in conclusion, D is definitely guilty, and A, B, and C are in doubt.Wait, but hold on. Let me check if A is definitely guilty or not.If A is guilty, then C is guilty, which makes D guilty. If A is not guilty, D is guilty. So, D is guilty either way.But is there a way that A must be guilty? Or can A be innocent?From the statements, if A is innocent, D is guilty. There's no contradiction in that. So, A can be innocent, so A is in doubt.Similarly, if A is guilty, that's also possible. So, A is in doubt.So, yeah, only D is definitely guilty.Therefore, the answer is D is definitely guilty, and A, B, and C are in doubt.**Final Answer**The defendant definitely guilty is boxed{D}.

answer：Okay, so I have this expression: [(2^{1004} + 5^{1005})^2 - (2^{1004} - 5^{1005})^2]. The problem says that this equals (k cdot 10^{1004}) for some positive integer (k), and I need to find (k). Hmm, let me think about how to approach this.First, I remember that there's a formula for the difference of squares: (a^2 - b^2 = (a - b)(a + b)). Maybe I can apply that here. Let me set (a = 2^{1004} + 5^{1005}) and (b = 2^{1004} - 5^{1005}). Then, the expression becomes:[(a)^2 - (b)^2 = (a - b)(a + b)]So, let's compute (a - b) and (a + b).Starting with (a - b):(a - b = (2^{1004} + 5^{1005}) - (2^{1004} - 5^{1005}))Simplify this:(2^{1004} - 2^{1004} + 5^{1005} + 5^{1005} = 0 + 2 cdot 5^{1005} = 2 cdot 5^{1005})Okay, so (a - b = 2 cdot 5^{1005}).Now, let's compute (a + b):(a + b = (2^{1004} + 5^{1005}) + (2^{1004} - 5^{1005}))Simplify this:(2^{1004} + 2^{1004} + 5^{1005} - 5^{1005} = 2 cdot 2^{1004} + 0 = 2^{1005})So, (a + b = 2^{1005}).Putting it all together, the original expression is:[(a)^2 - (b)^2 = (a - b)(a + b) = (2 cdot 5^{1005})(2^{1005})]Let me compute this product:First, multiply the constants: 2 and 2^{1005}.Wait, hold on, actually, (2 cdot 5^{1005}) times (2^{1005}) is:(2 cdot 2^{1005} cdot 5^{1005})Simplify the powers of 2:(2^{1} cdot 2^{1005} = 2^{1 + 1005} = 2^{1006})So now, the expression becomes:(2^{1006} cdot 5^{1005})Hmm, I need to express this as (k cdot 10^{1004}). Let me see how to manipulate this.First, note that (10^{1004} = (2 cdot 5)^{1004} = 2^{1004} cdot 5^{1004}).So, (k cdot 10^{1004} = k cdot 2^{1004} cdot 5^{1004}).Looking back at my expression, (2^{1006} cdot 5^{1005}), I can factor out (2^{1004} cdot 5^{1004}):(2^{1006} cdot 5^{1005} = 2^{1004} cdot 2^{2} cdot 5^{1004} cdot 5^{1})Which simplifies to:(2^{1004} cdot 5^{1004} cdot 2^{2} cdot 5^{1})That is:((2^{1004} cdot 5^{1004}) cdot (2^2 cdot 5))Which is:(10^{1004} cdot (4 cdot 5))Compute (4 cdot 5):(4 cdot 5 = 20)So, the expression becomes:(20 cdot 10^{1004})Therefore, (k cdot 10^{1004} = 20 cdot 10^{1004}), which implies that (k = 20).Wait, let me double-check my steps to make sure I didn't make a mistake.1. Applied the difference of squares correctly: yes, (a^2 - b^2 = (a - b)(a + b)).2. Calculated (a - b) as (2 cdot 5^{1005}): correct, because the (2^{1004}) terms cancel, and the (5^{1005}) terms add.3. Calculated (a + b) as (2^{1005}): correct, because the (5^{1005}) terms cancel, and the (2^{1004}) terms add.4. Multiplied (2 cdot 5^{1005}) by (2^{1005}): yes, which gave (2^{1006} cdot 5^{1005}).5. Expressed (10^{1004}) as (2^{1004} cdot 5^{1004}): correct.6. Factored out (2^{1004} cdot 5^{1004}) from (2^{1006} cdot 5^{1005}): correct, which leaves (2^2 cdot 5^1 = 4 cdot 5 = 20).So, all steps seem solid. Therefore, (k = 20).**Final Answer**boxed{20}

answer：Okay, so I have this problem here: Given that a is a real number, and (1 + ai)i equals 3 + i. I need to find the value of a. Hmm, let me think about how to approach this.First, I remember that i is the imaginary unit, where i squared is equal to -1. So, when dealing with complex numbers, I can use algebraic rules similar to real numbers, but I have to remember that i² = -1. That should help me simplify the expression.The expression given is (1 + ai)i. I need to multiply this out. Let me write that down step by step. So, I have:(1 + ai) * iI can distribute the i across both terms inside the parentheses. That means I'll multiply 1 by i and ai by i. Let's do that:1 * i + ai * iSimplifying each term, 1 * i is just i, and ai * i is a * i². So now, the expression becomes:i + a * i²But wait, I know that i² is equal to -1. So, substituting that in, the expression becomes:i + a * (-1)Which simplifies to:i - aSo, putting it all together, (1 + ai)i simplifies to i - a.Now, according to the problem, this expression equals 3 + i. So, I can set up the equation:i - a = 3 + iHmm, okay. Now, I need to solve for a. Let me write that equation again:i - a = 3 + iI can subtract i from both sides to get rid of the imaginary unit on both sides. Let me do that:i - a - i = 3 + i - iSimplifying both sides, on the left side, i - i cancels out, leaving -a. On the right side, i - i also cancels out, leaving 3. So now, the equation is:-a = 3To solve for a, I can multiply both sides by -1:(-a) * (-1) = 3 * (-1)Which simplifies to:a = -3Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the beginning:(1 + ai)i = 3 + iMultiply out the left side:1*i + ai*i = i + a*i²Since i² = -1, that becomes:i + a*(-1) = i - aSet equal to 3 + i:i - a = 3 + iSubtract i from both sides:-a = 3Multiply both sides by -1:a = -3Yep, that seems correct. So, the value of a is -3.Just to make sure, let me plug a = -3 back into the original expression and see if it equals 3 + i.Original expression: (1 + ai)iSubstitute a = -3:(1 + (-3)i)i = (1 - 3i)iMultiply it out:1*i - 3i*i = i - 3i²Again, since i² = -1:i - 3*(-1) = i + 3Which is 3 + i, which matches the right side of the equation. So, that checks out.Therefore, I'm confident that a is indeed -3.**Final Answer**The value of ( a ) is boxed{-3}.

answer：Okay, so I have this problem where 100 people were surveyed. Out of these, 87 said they liked Mozart and 70 said they liked Bach. I need to find the minimum number of people who could have said they liked both Mozart and Bach. Hmm, okay. Let me think about how to approach this.First, I remember something about sets and maybe using a Venn diagram to visualize this. So, if I consider the total number of people surveyed as the universal set, which is 100. Then, the number of people who like Mozart is 87, and the number who like Bach is 70. I need to find the overlap between these two groups, right? The overlap is the number of people who like both Mozart and Bach.Wait, but the question is asking for the minimum number of people who could have liked both. So, I think this is a problem about the intersection of two sets. I recall that the formula for the union of two sets is |A ∪ B| = |A| + |B| - |A ∩ B|. Maybe I can use this formula to find the intersection.Let me write that down:|A ∪ B| = |A| + |B| - |A ∩ B|Where:- |A| is the number of people who like Mozart, which is 87.- |B| is the number of people who like Bach, which is 70.- |A ∩ B| is the number of people who like both, which is what we're trying to find.- |A ∪ B| is the number of people who like either Mozart or Bach or both.But since the total number of people surveyed is 100, the maximum possible value for |A ∪ B| is 100. So, |A ∪ B| ≤ 100.Plugging the numbers into the formula:|A ∪ B| = 87 + 70 - |A ∩ B|Which simplifies to:|A ∪ B| = 157 - |A ∩ B|But we know that |A ∪ B| can't be more than 100, so:157 - |A ∩ B| ≤ 100Let me solve for |A ∩ B|:157 - |A ∩ B| ≤ 100 Subtract 157 from both sides: -|A ∩ B| ≤ -57 Multiply both sides by -1 (and reverse the inequality): |A ∩ B| ≥ 57So, the number of people who like both Mozart and Bach must be at least 57. That means the minimum number is 57.Wait, let me make sure I didn't make a mistake here. So, if 87 people like Mozart and 70 like Bach, adding those together gives 157. But since only 100 people were surveyed, the overlap must account for the excess. The excess is 157 - 100 = 57. So, that makes sense. The overlap can't be less than 57 because otherwise, the total number of people surveyed would have to be more than 100, which it isn't.Let me think of it another way. If as few people as possible liked both, then we want to maximize the number of people who liked only Mozart or only Bach. But since the total number of people is fixed at 100, the overlap can't be less than the sum of the two groups minus the total.Yeah, that seems right. So, the minimum number of people who liked both is 57.**Final Answer**The minimum number of people who could have said they liked both Mozart and Bach is boxed{57}.

answer：Okay, so I've got this problem here about shifting a sine function and figuring out the maximum value of ω such that the shifted function is increasing on a certain interval. Let me try to unpack this step by step.First, the original function is f(x) = 2 sin(2ωx - π/3), and ω is greater than 0. They say the graph is shifted to the left by π/(6ω) units to get y = g(x). Then, we need to find the maximum ω such that g(x) is increasing on [0, π/4].Alright, so shifting a function left by a units means replacing x with x + a in the function. So, for g(x), which is f shifted left by π/(6ω), it should be:g(x) = f(x + π/(6ω)) = 2 sin[2ω(x + π/(6ω)) - π/3]Let me simplify the argument inside the sine function:2ω(x + π/(6ω)) = 2ωx + 2ω*(π/(6ω)) = 2ωx + π/3So, the argument becomes 2ωx + π/3 - π/3 = 2ωx. Wait, that simplifies nicely!So, g(x) = 2 sin(2ωx). Hmm, that's interesting. So, shifting f(x) left by π/(6ω) units cancels out the phase shift in f(x). So, g(x) is just a sine function with amplitude 2 and frequency 2ω.Now, we need to find the maximum ω such that g(x) is increasing on [0, π/4]. So, let's recall that for a function to be increasing on an interval, its derivative should be non-negative on that interval.So, let's compute the derivative of g(x):g'(x) = d/dx [2 sin(2ωx)] = 2 * 2ω cos(2ωx) = 4ω cos(2ωx)So, g'(x) = 4ω cos(2ωx). We need this to be non-negative on [0, π/4]. Since ω > 0, 4ω is positive, so the sign of g'(x) depends on cos(2ωx).Therefore, for g'(x) ≥ 0 on [0, π/4], we need cos(2ωx) ≥ 0 for all x in [0, π/4].So, the question reduces to finding the maximum ω such that cos(2ωx) ≥ 0 for all x in [0, π/4].Let me think about the cosine function. Cosine is non-negative in the intervals where its argument is between -π/2 + 2πk and π/2 + 2πk for integers k.But since 2ωx is increasing as x increases (because ω > 0), the maximum value of 2ωx on [0, π/4] is 2ω*(π/4) = ωπ/2.So, to ensure that cos(2ωx) is non-negative for all x in [0, π/4], the maximum argument, which is ωπ/2, must be less than or equal to π/2. Because beyond π/2, cosine becomes negative.Wait, but is that the case? Let me think again.If the maximum argument is ωπ/2, then for cos(2ωx) to be non-negative throughout [0, π/4], the maximum argument must be less than or equal to π/2, because cosine starts to decrease after 0 and becomes negative at π/2.But actually, cosine is positive from -π/2 to π/2, but since 2ωx is positive (as ω > 0 and x ≥ 0), we only need to consider the interval from 0 to ωπ/2.So, to ensure that cos(2ωx) is non-negative on [0, π/4], the maximum value of 2ωx, which is ωπ/2, must be less than or equal to π/2.So, ωπ/2 ≤ π/2Divide both sides by π/2:ω ≤ 1Hmm, so ω ≤ 1. So, the maximum value of ω is 1.Wait, but let me verify this. If ω = 1, then g(x) = 2 sin(2x). Let's compute its derivative: g'(x) = 4 cos(2x). On [0, π/4], 2x goes from 0 to π/2. Cosine is positive on [0, π/2), so at x = π/4, 2x = π/2, and cos(π/2) = 0. So, the derivative is non-negative on [0, π/4], which satisfies the condition.If ω were greater than 1, say ω = 1 + ε for some ε > 0, then the maximum argument would be (1 + ε)π/2, which is greater than π/2. So, at x = π/4, 2ωx = (1 + ε)π/2, which is greater than π/2, and cosine of that would be negative. Therefore, the derivative would be negative at x = π/4, meaning the function is decreasing there, which violates the condition that it's increasing on the entire interval.Hence, ω must be at most 1.Wait, but let me think again. Is there a possibility that even if the maximum argument is beyond π/2, the function could still be increasing on [0, π/4] if the argument doesn't reach π/2 within that interval? Wait, no, because as x increases, 2ωx increases, so if ω is larger, 2ωx increases faster. So, if ω is too large, 2ωx could surpass π/2 before x reaches π/4, making cos(2ωx) negative somewhere in [0, π/4], which would make the derivative negative there.But wait, let's compute the point where cos(2ωx) = 0. That happens when 2ωx = π/2, so x = π/(4ω). So, if π/(4ω) is less than π/4, that would mean ω > 1. So, if ω > 1, then the point where the derivative becomes zero is at x = π/(4ω) < π/4. So, before x = π/4, the derivative becomes zero and then negative. Therefore, the function would start decreasing before x = π/4, which is not allowed.Hence, to ensure that the derivative doesn't become zero before x = π/4, we need π/(4ω) ≥ π/4, which simplifies to ω ≤ 1.Therefore, the maximum value of ω is 1.Wait, but let me check for ω = 1. At x = π/4, 2ωx = 2*(1)*(π/4) = π/2, so cos(π/2) = 0, so the derivative is zero at x = π/4. So, is the function still considered increasing on [0, π/4] if the derivative is zero at the endpoint?I think in calculus, a function is considered increasing on an interval if its derivative is non-negative throughout the interval. So, if the derivative is zero at the endpoint, it's still acceptable because it's non-negative everywhere on the interval, including the endpoints.Therefore, ω = 1 is acceptable.But let me think if there's another way to interpret this. Maybe considering the function's behavior beyond just the derivative. For example, if the function is increasing on [0, π/4], it doesn't necessarily have to be strictly increasing, just non-decreasing. So, having the derivative zero at the endpoint is okay.Alternatively, if we consider the function's monotonicity, if the derivative is non-negative on the interval, then the function is non-decreasing, which is what we need.So, yes, ω = 1 is the maximum value.Wait, but let me also think about the original function f(x) = 2 sin(2ωx - π/3). When we shift it left by π/(6ω), we get g(x) = 2 sin(2ωx). So, that makes sense.But just to make sure, let's plug ω = 1 into g(x). Then, g(x) = 2 sin(2x). The derivative is 4 cos(2x). On [0, π/4], 2x goes from 0 to π/2. Cosine is positive in [0, π/2), and zero at π/2. So, the derivative is non-negative throughout [0, π/4], so the function is increasing.If ω were slightly more than 1, say ω = 1.1, then g(x) = 2 sin(2.2x). The derivative is 4.4 cos(2.2x). The maximum argument at x = π/4 is 2.2*(π/4) ≈ 1.727 radians, which is more than π/2 (≈1.5708). So, cos(1.727) is negative, meaning the derivative is negative at x = π/4, which means the function is decreasing there, violating the condition.Hence, ω cannot be more than 1.Therefore, the maximum value of ω is 1.**Final Answer**The maximum value of ω is boxed{1}.

answer：Okay, so I have this problem where I need to find the number of ordered triples (a, b, c) of positive integers such that the least common multiples (LCMs) satisfy [a, b] = 1000, [b, c] = 2000, and [c, a] = 2000. Hmm, LCMs can sometimes be tricky, but I think breaking them down into their prime factors might help. Let me try that.First, I should factorize 1000 and 2000 into their prime components. 1000 is equal to 10^3, which is (2*5)^3, so that's 2^3 * 5^3. 2000 is 2^4 * 5^3 because 2000 is 2^4 * 5^3. So, both 1000 and 2000 have the same power of 5, which is 5^3, but 2000 has a higher power of 2, which is 2^4 compared to 1000's 2^3.Since LCMs are involved, I know that for two numbers, the LCM takes the highest power of each prime present in the numbers. So, for example, if I have two numbers x and y, [x, y] will have for each prime the maximum exponent between x and y.Given that, I can model each of a, b, c in terms of their exponents for primes 2 and 5 because 1000 and 2000 only involve these primes. So, I can represent a, b, c as:a = 2^{a2} * 5^{a5}b = 2^{b2} * 5^{b5}c = 2^{c2} * 5^{c5}Where a2, a5, b2, b5, c2, c5 are non-negative integers representing the exponents of primes 2 and 5 in each number.Now, the LCM conditions translate to:1. [a, b] = 1000 = 2^3 * 5^3 So, max(a2, b2) = 3 and max(a5, b5) = 3.2. [b, c] = 2000 = 2^4 * 5^3 So, max(b2, c2) = 4 and max(b5, c5) = 3.3. [c, a] = 2000 = 2^4 * 5^3 So, max(c2, a2) = 4 and max(c5, a5) = 3.So, now I can handle the exponents separately for primes 2 and 5.Let me first handle the exponents of 2. Let me denote:For prime 2:- a2: exponent of 2 in a- b2: exponent of 2 in b- c2: exponent of 2 in cWe have the following conditions:1. max(a2, b2) = 32. max(b2, c2) = 43. max(c2, a2) = 4Similarly, for prime 5:- a5: exponent of 5 in a- b5: exponent of 5 in b- c5: exponent of 5 in cConditions:1. max(a5, b5) = 32. max(b5, c5) = 33. max(c5, a5) = 3So, I can handle the exponents for 2 and 5 separately and then multiply the number of possibilities for each prime since they are independent.Let me start with prime 2.**Handling Prime 2:**We have:1. max(a2, b2) = 32. max(b2, c2) = 43. max(c2, a2) = 4I need to find the number of triples (a2, b2, c2) satisfying these conditions.First, let's note that from condition 2: max(b2, c2) = 4. So, at least one of b2 or c2 must be 4.Similarly, from condition 3: max(c2, a2) = 4. So, at least one of c2 or a2 must be 4.But from condition 1: max(a2, b2) = 3. So, both a2 and b2 must be ≤ 3, and at least one of them is 3.Wait, hold on. If max(a2, b2) = 3, then both a2 and b2 are ≤ 3, and at least one is 3.But from condition 2: max(b2, c2) = 4. Since b2 is ≤ 3, c2 must be 4.Similarly, from condition 3: max(c2, a2) = 4. Since c2 is 4, this condition is automatically satisfied regardless of a2.So, let's summarize:From condition 1: a2 ≤ 3, b2 ≤ 3, and at least one of a2 or b2 is 3.From condition 2: Since b2 ≤ 3, c2 must be 4.From condition 3: Since c2 is 4, this condition is satisfied.Therefore, c2 must be 4, and a2 and b2 are such that max(a2, b2) = 3.So, the number of possible (a2, b2) pairs is equal to the number of pairs where a2 ≤ 3, b2 ≤ 3, and at least one of a2 or b2 is 3.Let me compute that.For a2, possible values: 0,1,2,3Similarly for b2: 0,1,2,3Total pairs without any restriction: 4*4=16Number of pairs where both a2 and b2 are ≤2: 3*3=9Therefore, number of pairs where at least one is 3: 16 - 9 = 7.Wait, hold on: 4*4 is 16, and subtracting the 3*3=9 where both are ≤2 gives 7 pairs where at least one is 3.So, there are 7 possible (a2, b2) pairs.Since c2 is fixed at 4, the number of triples (a2, b2, c2) is 7.Wait, but let me double-check.Wait, if a2 can be from 0 to 3, and b2 can be from 0 to 3, with at least one being 3.So, the number of such pairs is:Number of a2=3: 4 (since b2 can be 0,1,2,3)Number of b2=3: 4 (since a2 can be 0,1,2,3)But this counts the case where both a2=3 and b2=3 twice, so we subtract 1.So, total is 4 + 4 -1 =7. Yes, that's correct.So, 7 possibilities for (a2, b2, c2).**Handling Prime 5:**Now, moving on to prime 5.We have:1. max(a5, b5) = 32. max(b5, c5) = 33. max(c5, a5) = 3So, all three maxima must be 3. So, for each pair among (a5, b5), (b5, c5), (c5, a5), the maximum must be 3.This implies that in each pair, at least one must be 3.Therefore, for each of a5, b5, c5, they can be at most 3, and in each pair, at least one is 3.So, let's think about the constraints:- For a5: Since max(a5, b5)=3 and max(a5, c5)=3, a5 can be anything from 0 to 3, but if a5 is less than 3, then both b5 and c5 must be 3.Similarly, for b5: Since max(a5, b5)=3 and max(b5, c5)=3, if b5 is less than 3, then both a5 and c5 must be 3.Similarly, for c5: Since max(b5, c5)=3 and max(c5, a5)=3, if c5 is less than 3, then both a5 and b5 must be 3.So, let's consider the possible cases.Case 1: All of a5, b5, c5 are 3.Case 2: Exactly two of them are 3, and the third is less than 3.Case 3: Exactly one of them is 3, and the other two are less than 3.But wait, let's check if Case 3 is possible.Suppose only one of them is 3, say a5=3, and b5, c5 <3.But then, max(b5, c5) must be 3, but both are less than 3, so that's a contradiction. Similarly, if only b5=3, then max(a5, c5) must be 3, but both a5 and c5 are less than 3, which is a contradiction. Similarly, if only c5=3, then max(a5, b5) must be 3, but both a5 and b5 are less than 3, which is a contradiction.Therefore, Case 3 is impossible.Similarly, let's check Case 2: Exactly two of them are 3, the third is less than 3.Suppose a5=3, b5=3, c5 <3.Then, max(a5, b5)=3, which is okay.max(b5, c5)=3, since b5=3.max(c5, a5)=3, since a5=3.So, that's okay.Similarly, if a5=3, c5=3, b5 <3.Then, max(a5, b5)=3, since a5=3.max(b5, c5)=3, since c5=3.max(c5, a5)=3, which is okay.Similarly, if b5=3, c5=3, a5 <3.Then, max(a5, b5)=3, since b5=3.max(b5, c5)=3, since both are 3.max(c5, a5)=3, since c5=3.So, all these are okay.So, in Case 2, we have three subcases:- a5=3, b5=3, c5 <3- a5=3, c5=3, b5 <3- b5=3, c5=3, a5 <3Each of these subcases is valid.Case 1: All three are 3.Case 2: Exactly two are 3, and the third is less than 3.Case 3: Impossible.So, let's compute the number of triples for each case.Case 1: All three are 3.Only 1 possibility: a5=3, b5=3, c5=3.Case 2: Exactly two are 3, the third is less than 3.For each subcase:Subcase 1: a5=3, b5=3, c5 <3.c5 can be 0,1,2. So, 3 possibilities.Subcase 2: a5=3, c5=3, b5 <3.b5 can be 0,1,2. So, 3 possibilities.Subcase 3: b5=3, c5=3, a5 <3.a5 can be 0,1,2. So, 3 possibilities.Total for Case 2: 3 + 3 + 3 =9.Case 3: Impossible, so 0.Therefore, total number of triples (a5, b5, c5) is 1 + 9 =10.Wait, is that correct?Wait, let me think again.Wait, in Case 2, each subcase is distinct? For example, in Subcase 1, a5=3, b5=3, c5 <3. Similarly, Subcase 2 is a5=3, c5=3, b5 <3. These are different because in Subcase 1, b5=3, while in Subcase 2, b5 <3. So, yes, they are distinct.Similarly, Subcase 3 is when a5 <3, which is different from the other two.So, total 9 possibilities for Case 2 and 1 for Case 1, so 10 in total.Wait, but let me think about whether these are all unique.Suppose in Subcase 1: a5=3, b5=3, c5=0,1,2.Similarly, in Subcase 2: a5=3, c5=3, b5=0,1,2.Similarly, Subcase 3: b5=3, c5=3, a5=0,1,2.So, these are all unique because in each subcase, the position of the non-3 exponent is different.Therefore, total 9 +1=10.So, 10 possibilities for (a5, b5, c5).**Combining Both Primes:**Since the exponents for 2 and 5 are independent, the total number of ordered triples (a, b, c) is the product of the number of possibilities for each prime.We have 7 possibilities for the exponents of 2 and 10 possibilities for the exponents of 5.Therefore, total number of triples is 7 * 10 =70.Wait, is that correct?Wait, let me make sure.Each exponent combination for 2 and 5 can be combined independently, so yes, the total number is multiplicative.So, 7 *10=70.Therefore, the number of ordered triples is 70.But wait, let me double-check my reasoning for prime 2.Earlier, I concluded that c2 must be 4 because from condition 2: max(b2, c2)=4, and since b2 ≤3, c2 must be 4.Similarly, from condition 3: max(c2, a2)=4, which is satisfied since c2=4.And from condition 1: max(a2, b2)=3, so a2 and b2 are ≤3, with at least one being 3.So, the number of (a2, b2) pairs is 7, as computed earlier.So, that seems correct.Similarly, for prime 5, the number of triples is 10, as computed.Therefore, the total number is 7*10=70.Hmm, that seems correct.But let me think again.Wait, another way to think about it is to model each exponent separately.For prime 2:We have c2=4, and a2 and b2 are such that max(a2, b2)=3.So, a2 can be from 0 to 3, b2 can be from 0 to 3, with at least one being 3.Which is 7 possibilities.For prime 5:We have all three exponents a5, b5, c5 must be such that in every pair, at least one is 3.Which gives 10 possibilities.Therefore, 7*10=70.Yes, that seems consistent.So, I think 70 is the correct answer.**Final Answer**The number of ordered triples is boxed{70}.