perplexus.info :: Numbers : lcm and gcd crossed square difference puzzle

Home > Numbers

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

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

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

Please log in:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info