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Positive integers x and y satisfy x < y. What is the value of y?
(1) The greatest common divisor of x and y is 6, and their least common multiple is 90.
(2) y − x is a multiple of 7.
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Correct answer
C
Use gcd(x, y) × lcm(x, y) = x × y. With gcd = 6 and lcm = 90, x × y = 540, and both x, y are multiples of 6 with x < y.
Statement (1): write x = 6m, y = 6n with gcd(m, n) = 1; then lcm = 6mn = 90 ⇒ mn = 15. co-prime pairs with m < n: (1, 15) → (x, y) = (6, 90); (3, 5) → (x, y) = (18, 30). both have gcd 6 and lcm 90, so y = 90 or y = 30. not sufficient.
Statement (2): y − x a multiple of 7 leaves both free. not sufficient.
together: test each candidate from (1) against (2). for (6, 90): y − x = 84 = 7 × 12 (a multiple of 7). for (18, 30): y − x = 12, NOT a multiple of 7. only (6, 90) survives both statements, so y = 90 uniquely. sufficient together; neither alone.
the gcd-lcm-product-not-unique trap plus the tie-breaker discrimination. solvers compute x × y = 540 and assume the pair is pinned, choosing A; but mn = 15 splits into two co-prime pairs (1, 15) and (3, 5), so (1) alone is genuinely ambiguous. (2) is the surgical tie-breaker: only one candidate's difference is a multiple of 7.
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