Determine all pairs (a,b) of positive integers that satisfy this equation:
lcm(a,b)2 - gcd(a,b)2 = 48
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Let a=f*g and b=g*h, such that g is the gcd and f is coprime to h.
Then the lcm is f*g*h.
Then the equation becomes (f*g*h)^2 - g^2 = 48.
This factors into ((f*h)^2 - 1) * g^2 = 48.
So we need a factorization of 48 into a perfect square g^2 and one less than a perfect square (f*h)^2 - 1. The perfect square factors of 48 are 1, 4, and 16.
48/1=7^2-1 is good. 48/16=2^2-1 is good. But 48/4=12 is not one less than a perfect square.
At this point we have only two possible cases: g=1 and f*h=7, or g=4 and f*h=2.
Case g=1 and f*h=7. Then f and h can only 7 and 1. Then a and b are 7 and 1. lcm(7,1)=7 and gcd(7,1)=1 and 7^2-1^2=48. So (a,b)=(1,7) or (7,1) are a couple of solutions.
Case g=4 and f*h=2. Then f and h can only be 2 and 1. Then a and b are 8 and 4. lcm(8,4)=8 and gcd(8,4)=4 and 8^2-4^2=48. So (a,b)=(4,8) or (8,4) are a couple of solutions.
All pairs (a,b) of positive integers that satisfy the given equation are (1,7), (7,1), (4,8), or (8,4).
Analytic Solution
