<p>Let <img alt="$A$" height="14" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="15" /> be the set of positive integer divisors of <img alt="$2025$" height="14" src="https://latex.artofproblemsolving.com/7/8/d/78db23e86da8fe319c63bdb5eaf6d3fd8fefa17b.png" width="38" />. Let <img alt="$B$" height="14" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="17" /> be a randomly selected subset of <img alt="$A$" height="14" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="15" />. The probability that <img alt="$B$" height="14" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="17" /> is a nonempty set with the property that the least common multiple of its elements is <img alt="$2025$" height="14" src="https://latex.artofproblemsolving.com/7/8/d/78db23e86da8fe319c63bdb5eaf6d3fd8fefa17b.png" width="38" /> is <img alt="$\frac{m}{n}$" height="34" src="https://latex.artofproblemsolving.com/2/7/f/27f583697295fc6f723862c13228ee55107c33ca.png" width="22" />, where <img alt="$m$" height="10" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="18" /> and <img alt="$n$" height="10" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="13" /> are relatively prime positive integers. Find <img alt="$m+n$" height="14" src="https://latex.artofproblemsolving.com/0/2/d/02dbab3601fa6cce490d54263017048f69e1a35d.png" width="51" />.</p>

Question No: 454

Let $A$ be the set of positive integer divisors of $2025$ . Let $B$ be a randomly selected subset of $A$ . The probability that $B$ is a nonempty set with the property that the least common multiple of its elements is $2025$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

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Tags: Combinatorics Number Theory Probability

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