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# Counting Divisors

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Some positive integers have exactly four positive factors. For example, 35 has only 1, 5, 7 and 35 as its factors. What is the sum of the smallest five positive integers that each have exactly four positive factors?

Apr 5, 2021

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If they have four factors they can be of the form $p_1\cdot p_2$ or $p^3$ for all $p_1,p_2,p$ prime.

Let's do $p_1\cdot p_2$ first. We have $2\cdot 3=6, 2\cdot 5=10, 2\cdot 7=14, 2\cdot 11=22,$ etc. if $2$ is the first. Otherwise we can have $3\cdot 5=15, 3\cdot 7=21$, etc. So the smallest $5$ are $6,10,14,15,21$.

Then for $p^3$ we can have $2^3=8$ and the rest are larger than $21$.

So the numbers are $6,8,10,14,15$. Sum them up.

Apr 5, 2021
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Thank you @thedudemanguyperson, it was correct!

RiemannIntegralzzz  Apr 5, 2021