<p>Is mn 60? Because that would make the product 15mn equal to 900, which is a perfect square.</p>
<p>hes right ^</p>
<p>Oh hell. 36 is a perfect square not 360. Damn :(</p>
<p>Is there a straight-forward way of solving this or can one only plug-in values?</p>
<p>IMO, the answer is 60 and the way of solving is:</p>
<p>for 15mn to be a perfect square , the value of mn can be any of the following format:</p>
<p>15 * ( a perfect square )</p>
<p>the least value of a perfect square other than 1 is 4. </p>
<p>So the answer is 15*4=60</p>
<p>There’s certainly a straight-forward method of solving the above problem. To solve it, you must implement number theory concepts.</p>
<p>First, a square number is one in which its prime factorization has even exponents. 15mn translates to 3^1 * 5^1 *mn. Thus, we are looking for a number that has 3^2 and 5^2 in its prime factorization (as this will yield the least square number). Therefore, mn = 3^1 * 5^1 *some square prime (because multiplying this by 15 gives us 3^2 * 5^2 in the prime factorization of the square number). mn cannot simply be 15, because it is specified as being greater than such. Since we are looking for the smallest square, mn is equal to 3^1 * 5^1 * 2^2 (2 being the smallest prime number that we can square), or 60. If we multiply this by 15, we get 2^2 * 3^2 * 5^2, or 900 (which is a perfect square, already evident by the prime factorization). </p>
<p>Therefore, mn is 60.</p>
<p>I hope this solution is intelligible to those unfamiliar with number theory. If anyone is confused, I’d be glad to clarify.</p>
<p>An appropriate screen name - Studious!</p>
<p>I hadn’t known of such an approach when I attacked the problem! Thanks for sharing :).</p>
<p>Happy to help, IceQube! I recommend checking out the Art of Problem Solving Number Theory book. It’s been a great boon to my understanding of problems suvh as this one.</p>
<p>The assistant to the doctor, who graduated in three years, has refused an offer of promotion. no error.</p>
<p>Damn, is it who? Ambiguity?</p>
<p>Although…a subordinate clause usually modifies the word directly before…hmph</p>
<p>I think it’s “who” too.</p>