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lcm and gcd crossed square difference puzzle (Posted on 2024-09-22)

Determine all pairs (a,b) of positive integers that satisfy this equation:

lcm(a,b)² - gcd(a,b)² = 48

No Solution Yet Submitted by K Sengupta

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Partial analytic solution

| Comment 2 of 4 |

I ran it up to 10000 and still found only the same 2 solutions as Charlie: {1,7} and {4,8}

Proof that {1,7} is the only solution in the special case that a and b are relatively prime:

Then lcm = ab and gcd = 1 so we have:

(ab)^2 - 1 = 48

(ab)^2 = 49

ab = 7

{a,b} can only be {1,7}

Posted by Larry on 2024-09-22 09:16:48

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