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.

$\begin{eqnarray}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}=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

\end{eqnarray}$ (Error compiling LaTeX. Unknown error_msg)

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.

Furthermore, if we rearrange $\textbf{(2)}$ , we can see that the number would be the sum of all its divisors other than itself, making it a perfect number. Checking the perfect numbers, we find that the second perfect number, $28$ , fits the required form and is indeed the solution.

2005 Alabama ARML TST (Problems)
Preceded by: Problem 12	Followed by: Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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2005 Alabama ARML TST Problems/Problem 13

Problem

Solution

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