DS数论难题

Jiamisnm

求助大佬们一道DS题！！！

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.
  

tankhuang

搬运外网解答：

Statement 1: x=12u, where u is an integer and x=8y+12.
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.

If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.

If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.

Since the GCD can be different values, INSUFFICIENT.

Statement 2: y=12z, where z is an integer and x=8y+12.
In other words, y is a multiple of 12.
Since we're looking for the GCD, view x in terms of its FACTORS.

If y=12, then x = 8(12) + 12 = 12(8+1) = 12*9.
The GCD of 12 and 12*9 is 12.

If y=24, then x = 8(24) + 12 = 12(8*2 + 1) = 12*17.
The GCD of 24 and 12*17 is 12.