2005 Alabama ARML TST Problems/Problem 13

Problem

There is one natural number with exactly 6 positive divisors, the sum of whose reciprocals is 2. Find that natural number.

Solution

Let the number be $n$ , and let $p_1$ and $p_2$ be primes. Therefore, one of the following is true:

$n=p_1^5$
$n=p_1p_2^2$

For the first one, the sum of the reciprocals of the divisors of $n$ is therefore $1+\dfrac{1}{p_1}+\dfrac{1}{p_1^2}+\dfrac{1}{p_1^3}+\dfrac{1}{p_1^4}+\dfrac{1}{p_1^5}$ . The smallest prime (2) makes that less than 2, and if $p_1$ gets bigger, then that expression gets smaller, so there is absolutely no way that $n=p_1^5$ . So the second case is true.

$1+\dfrac{1}{p_1}+\dfrac{1}{p_1p_2}+\dfrac{1}{p_1p_2^2}+\dfrac{1}{p_2}+\dfrac{1}{p_2^2}=2$

$\dfrac{p_1p_2^2+p_1p_2+p_1+p_2^2+p_2+1}{p_1p_2^2}=\dfrac{(p^2+p_2+1)(p_1+1)}{p_1p_2^2}=2$

$p_1p_2^2-p_1p_2-p_1=p_2^2+p_2+1$

$p_1(p_2^2-p_2-1)=p_2^2+p_2+1$

Therefore, $-p_1\equiv 1\bmod{p_2}$ . Now the only way that that is possible is when $p_2=2$ . Solving for $p_1$ , we get that $p_1=7$ . Checking, the sum of the reciprocals of the divisors of $\boxed{28}$ indeed sum to 2, and 28 does have 6 factors.

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2005 Alabama ARML TST Problems/Problem 13

Problem

Solution

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