abhi332 wrote:If a and b are consecutive positive integers, and ab = 30x is x a non-integer?
(1) a^2 is divisible by 21
(2) 35 is a factor of b^2
[spoiler]OA:C
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This is a bad question. I'll explain why at the bottom of my post, but first:
ab = 30x; is x a non-integer?
Does ab/30 = a (non) integer?
In order for the product ab to be divisible by 30, the product ab has to contain 2, 3, and 5 in its prime factorization. Because they are consecutive, we know exactly one of a and b is even; so, we know ab already has 2 in its prime factorization. The question becomes: does ab have 3 and 5 in its prime factorization?
(1) a^2 is divisible by 21;
In order for a^2/21 to be an integer, a has to have 3 and 7 in its prime factorization. But either of a or b may or may not be a multiple of 5...insufficient.
(2) 35 is a factor of b^2;
therefore, b has 5 and 7 in its prime factorization. But either of a or b may or may not be a multiple of 3...insufficient.
Together, you know that ab is a multiple of 3 and 5. Sufficient in combination.
HOWEVER, this question is flawed in design, as I remarked above. The information in the question stem contradicts the information in the combined statements: The question stem tells us that a and b are consecutive positive integers while the combined statements tell us that both a and b are multiples of 7. But two consecutive integers cannot be multiples of the same positive integer (unless that integer is 1). Without the word "consecutive", the question would be fine.
What is the source of this question?
