2021 AMC 10B Problem 12

Problem 12 of 25IntermediateNumber 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

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

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 the sum of the odd divisors. Therefore, the requested ratio is

1:14.

Thus, the answer is C .