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题库 / Prep2007E2-DS-80
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.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
题目要求x和y的最大公约数。条件一：x是12的倍数，所以y一定是3的倍数，但无法确定x和y的最大公约数（只能确定有公约数3，但并不知道是否最大）条件二：y是12的倍数，原式就可以变化成x=(8z+1)12 y=12z 所以最大公倍数是12，充分。选择B
(1) x = 12u, where u is an integer --> x=12ux=12u --> 12u=8y+1212u=8y+12 --> 3(u−1)=2y3(u−1)=2y --> the only thing we know from this is that 3 is a factor of y. Is it GCD of x and y? Not clear: if x=36x=36, then y=3y=3 and GCD(x,y)=3 GCD(x,y)=3
but if x=60x=60, then y=6y=6 and GCD(x,y)=6 --> two different answers. Not sufficient.
(2) y = 12z, where z is an integer --> y=12z --> x=8∗12z+12 --> x=12(8z+1).
So, we have y=12z and x=12(8z+1). Now, as z and 8z+1 do not share any common factor but 1 (8z and 8z+1 are consecutive integers and consecutive integers do not share any common factor 1. As 8z has all factors of z then z and 8z+1 also do not share any common factor but 1).
Thus, 12 must be GCD of x and y. Sufficient.
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