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Target question: What is the greatest common divisor of x and y?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
Given: x = 8y + 12
Statement 1: x = 12u, where u is an integer.
There are several pairs of values that satisfy the given conditions. Here are two:
Case a: x=36 and y=3, in which case the GCD of x and y is 3
Case b: x=60 and y=6, in which case the GCD of x and y is 6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: y = 12z, where z is an integer.
If y = 12z and x = 8y + 12, then we can replace y with 12z to get:
x = 8(12z) + 12, which means x = 96z + 12, which means x = 12(8z + 1) [if we factor]
So, what is the GCD of 12z and 12(8z + 1)?
Well, we can see that they both share 12 as a common divisor, but what about z and 8z+1?
Well, there's a nice rule that says: The GCD of n and kn+1 is always 1 (if n and k are positive integers)
So, the GCD of z and 8z+1 is 1, which means the GCD of 12z and 12(8z + 1) is 12.
This means that the GCD of x and y is 12
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = B
Cheers,
Brent
