Determine all pairs (a,b) of positive integers that satisfy this equation:
lcm(a,b)2 - gcd(a,b)2 = 48
This is essentially the same proof as provided by Brian Smith, only I didn't fully follow his set-up.
Let a=m*a1 and b=m*b1 where m is the greatest common factor of a and b.
Then lcm(a,b)=m*lcm(a1,b1) = m*a1*b1 and gcd(a,b)=m*gcd(a1,b1)=m. Applied to the problem, this means m^2 factors 48 and m=1,2,4.
m=1: Per Larry, a1*b1=7 and (a1,b1)=(a,b)=(1,7).
m=2: (a1*b1)^2=13, an impossibility.
m=4: (a1*b1)^2=4 so that (a1,b1)=(1,2) and (a,b)=(4,8).
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Posted by xdog
on 2024-09-22 13:20:34 |
solution
