SOLUTION: How many pairs of positive integers m,n satisfy 1/m + 4/n=1/12, where m is an odd integer less than 60?

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?
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!
We are given that such that is odd and .
Let us try and find out the value of in terms of
=>
=>
=>
=>
=>

So, we can say that
Possible odd values of such that are

| | | | | | | | | ||||||||||||||| ( integers)
check possible pairs:
,
,
, (irreducible)=> not an integer
same is if =>irreducible
,
, =>irreducible
, (irreducible)
, (irreducible)
, (irreducible)
, (irreducible)
, (irreducible)
, (irreducible)
and rest are irreducible

so, you have 3 pairs that satisfy given condition
,=>,
,=>,
, =>,

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

Tutor @MathLover1 has the right answers.

Here is a way to find them without trying all the odd integers less than 60.

In her solution, she shows algebraically that

Perform the division indicated by that expression:

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

The problem requires m to be an odd integer; that means m-12 is an odd integer.

The prime factorization of 576 is (2^6)(3^2).

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

So the only possible values of m-12 are 1, 3, and 9; that makes the possible values of m 13, 15, and 21.

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
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