2010 AMC 12B Problems/Problem 25

For every integer $n\ge2$ , let $\text{pow}(n)$ be the largest power of the largest prime tha divides $n$ . For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$ . What is the largest integer $m$ such that $2010^m$ divides

$\prod_{n=2}^{5300}\text{pow}(n)$ ?

$\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78$

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2010 AMC 12B Problems/Problem 25