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Bunuel
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I think this is a high-quality question and I agree with explanation.
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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did not get the line "for the integer x ..... x is 3^2 * 5^2" Could you pls elaborate on this?­
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tanishqgirotra
did not get the line "for the integer x ..... x is 3^2 * 5^2" Could you pls elaborate on this?­
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Since x is an integer, x^2 is the square of an integer, thus the powers of its primes must be even. For instance, if \(x = p^a*q^b*...\), then \(x^2 = p^{2a}*q^{2b}*...­\). Thus, the least value of \(x^2\) such that it is a multiple of \(3^3*5^3\) is \(3^4*5^4\).
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I like the solution - it’s helpful.
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I did not quite understand the solution. I didn't understand this part: "For the integer \(x\), the minimum value of \(x^2\) that can be a multiple of \(3^3*5^3\) is \(3^4*5^4\). Hence, the smallest possible value for \(x\) is \(3^2*5^2\). " Can you please explain? Maybe giving an example?
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micaelbalioni
I did not quite understand the solution. I didn't understand this part: "For the integer \(x\), the minimum value of \(x^2\) that can be a multiple of \(3^3*5^3\) is \(3^4*5^4\). Hence, the smallest possible value for \(x\) is \(3^2*5^2\). " Can you please explain? Maybe giving an example?

We want x^2 to be divisible by both 27 (which is 3^3) and 375 (which is 3 * 5^3). So x^2 must be divisible by the least common multiple of those, which is 3^3 * 5^3.

Now, what is the smallest possible value of x such that x^2 is divisible by 3^3 * 5^3?

To make sure x^2 includes at least 3^3 and 5^3, x must include at least 3^2 and 5^2. That way, x^2 = (3^2 * 5^2)^2 = 3^4 * 5^4, which covers 3^3 * 5^3.

So the smallest possible x is 3^2 * 5^2 = 225, and every x in the set T must be a multiple of 225.
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