2021 AMC 10B Spring Problem 12

Below is the professionally curated solution for Problem 12 of the 2021 AMC 10B Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10B Spring solutions, or check the answer key.

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Concepts:sum of factors prime factorization parity

Difficulty rating: 1140

12.

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 $N?$

$1 : 16$

$1 : 15$

$1 : 14$

$1 : 8$

$1 : 3$

Solution:

Using prime factorization, we get $N =2^3\cdot 3^5\cdot 5\cdot 7\cdot 17^2.$

If we have an odd divisor $x$ of $N,$ then $2x,4x,8x$ are divisors of $N,$ which has a combined sum of $14x.$ If we take the sum of every odd divisor, then the even divisors must have a sum which is $14$ times of the sum of the odd divisors.

Thus, the answer is C .

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