Number Prop

[artstudent]

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.

[Frankenstein]

Hi,
From(1):
if x= 36, y= 3. GCD of x and y is 3
if x= 60, y= 6. GCD of x and y is 6
So, it depends on the value of u
Not sufficient

From(2):
y = 12z
x = 8(12z) + 12 = 12(8z+1)
8z+1 when divided by z(>1) always gives remainder 1. So, z and 8z+1 do not have any common factors except one. That is z and 8z+1 are relatively prime. So, their GCD is 1
Hence, GCD of 12z and 12(8z+1) is 12.
Sufficient

Hence, B

[amit2k9]

a for x=36= 8*3+12, hence 36,3 have 3 as max divisor.
for x=60= 8*6+12, hence 60,6 have 6 as max divisor. Not sufficient.

b x= 12(8z+1). will always have 12 as the common factor.
also z and 8z+1 are for z=1,8z+1=9, for z=2, 8z+1=17. hence max divisor = 1 always.
Hence B it is.