[email protected] wrote:
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
Target question:
What is the greatest common divisor of x and y?
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
