How many positive integers $x$ with $x\mid 20^{20}$ have exactly $20$ divisors ?
2026-03-28 02:07:55.1774663675
Number of divisors of $ 20^{20} $ with exactly $20$ divisors
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Since $20^{20}=2^{40}5^{20}$, the general form of such a factor is $2^a5^b$ for non-negative integers $a,\,b$ with $(a+1)(b+1)=20$. Note in particular the ordered pair $(a,\,b)$ is what matters, not the unordered pair $\{a,\,b\}$. There are exactly as many of these as there are factors of $20=2^25$, i.e. $3\times2=6$.