The integer n is equal to k^2 for some integer k. If n is divisible by 24 and by 10, what is the smallest possible positive value of n ?

To get this question right quickly, you should really read this.

If n is a perfect square divisible by 24 and 10, then it needs to have all the prime factors of 24 and 10 in pairs. The prime factorization of 24 is 2^3 \times 3 and the prime factorization of 10 is 2\times 5.

Conveniently, the three 2s in 24 and the one 2 in 10 make two pairs of 2s, but in order to make a perfect square with all those factors, you’re going to need to add another 3 and another 5. So the smallest n could be, if it has to be a perfect square, is 2^4 \times 3^2 \times 5^2=3600.

For more discussion of this question, check out this post on the Tumblr Q&A.

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