Hi i need help with a problem i've been stuck on.
The natural numbers a and b are such that
(a + 1)/b + (b +1)/a
is an integer. Show that the greatest common divisor of a and b is not greater than sqrt(a + b).
I'm really stuck on this and I keep going round in circles. cannot prove anything.
any suggestions appreciated. Thank you.
well , if (a + 1)/b + (b +1)/a is a positive integer , we must have :
b divides a+1 and a divides b+1
let d be the greatest common divisor of a and b then d|a and d|b , hence , d| b+1 and d|b so d=1 .
we must now show that sqrt(a+b) is greater than 1 which is true since a and b (must be) are positive integers i.e a>=1 and b>=1 by summing we get sqrt(a+b) >=sqrt(2)>1 .