Very tricky question. Really took sometime to solve.
x = 8y+12. GCD(x,y)?
1. x = 12u
12u = 8y + 12, 12u is the multiple of 12 and we have a 12 constant. Therefore, 8y is also a multiple of 12.
12 is 2*2*3. Since in 8y has factors 2*2*2, to have 8y as a multiple of 12, then y must have a 3. But we are not sure what other factors y has got. Insufficient.
2. y = 12z
x = 8*12z + 12, (8*12z) is a multiple of 12 and we have a constant 12. Therefore x must be a multiple of 12.
Since we have 12 as a common divisor for x and 8y and x is 12 away from 8y, we cannot have a GCD of more than 12. So, 12 is GCD of x and y.
Sufficient.
Note: These concepts are brilliantly given in MGMAT: Number properties book. Its really worth to have a read at it.
