If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

Given: \(x=8y+12\).

(1) x = 12u, where u is an integer --> \(x=12u\) --> \(12u=8y+12\) --> \(3(u-1)=2y\) --> the only thing we know from this is that 3 is a factor of \(y\). Is it GCD of \(x\) and \(y\)? Not clear: if \(x=36\), then \(y=3\) and \(GCD(x,y)=3\) but if \(x=60\), then \(y=6\) and \(GCD(x,y)=6\) --> two different answers. Not sufficient.

(2) y = 12z, where z is an integer --> \(y=12z\) --> \(x=8*12z+12\) --> \(x=12(8z+1)\). So, we have \(y=12z\) and \(x=12(8z+1)\). Now, as \(z\) and \(8z+1\) do not share any common factor but 1 (8z and 8z+1 are consecutive integers and consecutive integers do not share any common factor 1. As 8z has all factors of z then z and 8z+1 also do not share any common factor but 1). Thus, 12 must be GCD of \(x\) and \(y\). Sufficient.

Answer: B.

Hope it's clear.

Can you please explain the Bunuel?

3(u-1)=2y

So, we know that 3 is a factor of 2y, but how can we conclude that 3 is a factor of y? Is it because 2 and 3 don't have any factor in common, except 1?

If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer.
(2) y = 12z, where z is an integer.