Number theory problem, fractions and gcd The problem says:If a and b are positive integers...

Emmy Dillon

Emmy Dillon

Answered

2022-06-26

Number theory problem, fractions and gcd
The problem says:
If a and b are positive integers such that a + 1 b + b + 1 a is an integer, then show that a + b gcd ( a , b )
Adding 2 a b a b to a + 1 b + b + 1 a yields that ab divides (a+b+1)(a+b), but I haven´t been able to continue from there.

Answer & Explanation

Carmelo Payne

Carmelo Payne

Expert

2022-06-27Added 25 answers

If gcd ( a , b ) = 1, then clearly a + b gcd ( a , b ). Now assume gcd ( a , b ) = d > 1. Then if a + 1 b + b + 1 a = n, then by your simplification, we have
( a + b + 1 ) ( a + b ) = ( n + 2 ) a b
Note that d 2 divides the righthand side, so it must also divide the lefthand side. Since d divides a and d divides b, d must divide a + b. Since a + b + 1 and a + b are coprime, and since d divides a + b, we must also have d 2 dividing a + b. In particular, a + b d 2 a + b d

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