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Number theory Difficulty 1.6 Junior Find the answer Canada

The positive divisors of 6 are 1, 2, 3, 6. The sum of the
positive divisors of 1184 is

Pick one

Solution

We start by finding the prime factors of 1184: $1184 = 2 \cdot 592 = 2^2 \cdot 296 = 2^3 \cdot 148 = 2^4 \cdot 74 = 2^5 \cdot 37$ The positive divisors of 1184
cannot contain prime factors other than 2 and 37, and cannot contain
more than 5 factors of 2 or 1 factor of 37.

Thus, the positive divisors are $1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184$ (The first five of these
divisors have 0 factors of 37 and 0 through 5 factors of 2, while the
last five have 1 factor 37 and 0 through 5 factors of 2.)

The sum, $S$ , of these divisors is
$\begin{align*} S & = 1 + 2 + 4 + 8 + 16 + 32 + 37 + 74 + 148 + 296 + 592 + 1184 \\ & = (1+2+4+8+16+32) + 37 \cdot (1+2+4+8+16+32) \\ & = (1+2+4+8+16+32)\cdot (1+37) \\ & = 63 \cdot 38 \\ & = 2394\end{align*}$

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.