2021 AMC 12B Problems/Problem 7

Problem

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 $N$ ?

$\textbf{(A)} ~1 : 16 \qquad\textbf{(B)} ~1 : 15 \qquad\textbf{(C)} ~1 : 14 \qquad\textbf{(D)} ~1 : 8 \qquad\textbf{(E)} ~1 : 3$

Solution

Prime factorize $N$ to get $N=2^{3} \cdot 3^{5} \cdot 5\cdot 7\cdot 17^{2}$ . For each odd divisor $n$ of $N$ , there exist even divisors $2n, 4n, 8n$ of $N$ , therefore the ratio is $1:(2+4+8)\rightarrow\boxed{\textbf{(C)}}$

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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All AMC 12 Problems and Solutions

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2021 AMC 12B Problems/Problem 7

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