(Adapted)(Olympiad of Moldova) Let $n = 2^{13}3^{11}5^7$. If $d\mid n^2$, $d < n$ and $d \nmid n$, find how many numbers $d$ exist.
2026-03-30 05:12:26.1774847546
How many positive integers $d$ satisfy the conditions?
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how many divisors of $n^2$ are less than $n$? they are exactly half of one less than the number of divisors. (pair up $d$ and $n^2/d$, the only one with no pair is $n$).
From these you must subtract all the proper divisors of $n$ and you have your answer. The final ingredient is the number of divisors formula which I'm sure you know.