document.write( "Question 1132291: How many pairs of positive integers m,n satisfy 1/m + 4/n=1/12, where m is an odd integer less than 60? \n" ); document.write( "

Algebra.Com's Answer #749310 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "Tutor @MathLover1 has the right answers.

\n" ); document.write( "Here is a way to find them without trying all the odd integers less than 60.

\n" ); document.write( "In her solution, she shows algebraically that

\n" ); document.write( " $\"n+=+48m%2F%28m-12%29\"$

\n" ); document.write( "Perform the division indicated by that expression:

\n" ); document.write( " $\"n+=+48m%2F%28m-12%29+=+%2848m-576%2B576%29%2F%28m-12%29+=+48+%2B+576%2F%28m-12%29\"$

\n" ); document.write( "In this form, 48 is an integer, and n has to be an integer; that means 576/(m-12) has to be an integer.

\n" ); document.write( "The problem requires m to be an odd integer; that means m-12 is an odd integer.

\n" ); document.write( "The prime factorization of 576 is (2^6)(3^2).

\n" ); document.write( "Therefore, for 576/(m-12) to be an integer, with m-12 odd, m-12 has to be a factor of 3^2.

\n" ); document.write( "So the only possible values of m-12 are 1, 3, and 9; that makes the possible values of m 13, 15, and 21.

\n" ); document.write( "m = 13 --> n = 48+576/1 = 48+576 = 624
\n" ); document.write( "m = 15 --> n = 48+576/3 = 48+192 = 240
\n" ); document.write( "m = 21 --> n = 48+576/9 = 48+64 = 112 \n" ); document.write( "

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