2021 AMC 12B Spring Problem 7

Attempt Problem 7 of the 2021 AMC 12B Spring below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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 $N?$

$1:16$

$1:15$

$1:14$

$1:8$

$1:3$

Factoring, $34=2\cdot 17,$ $63=3^2\cdot 7,$ and $270=2\cdot 3^3\cdot 5,$ so $N=2^3\cdot 3^5\cdot 5\cdot 7\cdot 17^2.$

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.