2021 AMC 12B Problem 7

Problem 7 of 25EasierNumber Theory

Let

N=34\cdot 34\cdot 63\cdot 270.

What is the ratio of the sum of the odd divisors of

N

to the sum of the even divisors of

N?

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Solution

Factoring,

34=2\cdot 17,

63=3^2\cdot 7,

and

270=2\cdot 3^3\cdot 5,

N=2^3\cdot 3^5\cdot 5\cdot 7\cdot 17^2.

Let

M

be the odd part

3^5\cdot 5\cdot 7\cdot 17^2.

The sum of all divisors is

(1+2+4+8)\,\sigma(M)

=15\,\sigma(M).

The odd divisors sum to

\sigma(M),

so the even divisors sum to

15\,\sigma(M)-\sigma(M)=14\,\sigma(M).

The ratio is

\sigma(M):14\,\sigma(M)=1:14.

Thus, the correct answer is C.