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
A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient
B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient
C) Both statements together are sufficient, but neither statement alone is sufficient
D) Each statement alone is sufficient
E) Statements (1) and (2) together are not sufficient
The answer is B, but I need to know why.
EDIT:
Ok, here is the solution
Statement (1) implies that x is a multiple of 12. If x = 36, then y = 36-12/8 = 3 and the greatest common divisor of x and y is 3. However, if x = 60, then y = 6 and the greatest common divisor of x and y is 6. Therefore, statement (1) alone is not sufficient. Statement (2) Implies that y is a multiple of 12 or that 12 is a divisor of y. Since x = 8y +12, it follows that 12 is a divisor of x, and thus 12 is a common divisor of x and y. Since 12 = x-8y, any common divisor of x and y must be a divisor of 12. Therefore no integer greater than 12 is a common divisor of x and y, and 12 is the greatest common divisor of x and y. Thus, statement (2) alone is sufficient.
1) x = 12u, where u is an integer
2) y = 12z, where z is an integer
A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient
B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient
C) Both statements together are sufficient, but neither statement alone is sufficient
D) Each statement alone is sufficient
E) Statements (1) and (2) together are not sufficient
The answer is B, but I need to know why.
EDIT:
Ok, here is the solution
Statement (1) implies that x is a multiple of 12. If x = 36, then y = 36-12/8 = 3 and the greatest common divisor of x and y is 3. However, if x = 60, then y = 6 and the greatest common divisor of x and y is 6. Therefore, statement (1) alone is not sufficient. Statement (2) Implies that y is a multiple of 12 or that 12 is a divisor of y. Since x = 8y +12, it follows that 12 is a divisor of x, and thus 12 is a common divisor of x and y. Since 12 = x-8y, any common divisor of x and y must be a divisor of 12. Therefore no integer greater than 12 is a common divisor of x and y, and 12 is the greatest common divisor of x and y. Thus, statement (2) alone is sufficient.
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