Let a be an integer, then there are integers X, Y such that aX+(a+1)Y=1.

Question

Find the smallest positive value of Y.

Answer 1

aX + aY + a = 1
a((X+Y+1)=1
a = 1/(X+Y+1)
The only way that a can be an integer is for
X+Y+1 = 1 , to get 1/1
X = -Y

so X and Y must be opposite integers, and of course the smallest positive integer possible for X is 1