Question 1132291
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Tutor @MathLover1 has the right answers.<br>
Here is a way to find them without trying all the odd integers less than 60.<br>
In her solution, she shows algebraically that<br>
{{{n = 48m/(m-12)}}}<br>
Perform the division indicated by that expression:<br>
{{{n = 48m/(m-12) = (48m-576+576)/(m-12) = 48 + 576/(m-12)}}}<br>
In this form, 48 is an integer, and n has to be an integer; that means 576/(m-12) has to be an integer.<br>
The problem requires m to be an odd integer; that means m-12 is an odd integer.<br>
The prime factorization of 576 is (2^6)(3^2).<br>
Therefore, for 576/(m-12) to be an integer, with m-12 odd, m-12 has to be a factor of 3^2.<br>
So the only possible values of m-12 are 1, 3, and 9; that makes the possible values of m 13, 15, and 21.<br>
m = 13  -->  n = 48+576/1 = 48+576 = 624
m = 15  -->  n = 48+576/3 = 48+192 = 240
m = 21  -->  n = 48+576/9 = 48+64 = 112