<p>A positive integer <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> has <img alt="$60$" height="12" src="https://latex.artofproblemsolving.com/3/0/2/302e99c1c7cd91a6ae371139e5142ccef5977dca.png" width="17" /> divisors and <img alt="$7n$" height="12" src="https://latex.artofproblemsolving.com/4/0/9/40905e7046ff302141d715350424107af2ee3d67.png" width="19" /> has <img alt="$80$" height="12" src="https://latex.artofproblemsolving.com/f/e/a/fea5fcc6d66685ee21f722efcd228eb44d38f262.png" width="17" /> divisors. What is the greatest integer <img alt="$k$" height="12" src="https://latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png" width="9" /> such that <img alt="$7^k$" height="15" src="https://latex.artofproblemsolving.com/d/9/c/d9c75b34bd42bc8596daa00e3f6ddd040a47b4ff.png" width="16" /> divides <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" />?</p>

Question No: 256

A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$ ?

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Tags: Algebra Probability

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