HMMT 二月 2003 · 冲刺赛 · 第 27 题

英文解析

27. The rational numbers x and y , when written in lowest terms, have denominators 60
    and 70, respectively. What is the smallest possible denominator of x + y ?
    Solution: 84
    Write x + y = a/ 60 + b/ 70 = (7 a + 6 b ) / 420. Since a is relatively prime to 60 and b isrelatively prime to 70, it follows that none of the primes 2 , 3 , 7 can divide 7 a + 6 b , so wewon’t be able to cancel any of these factors in the denominator. Thus, after reducingto lowest terms, the denominator will still be at least 2 · 3 · 7 = 84 (the product of 2
    the powers of 2 , 3, and 7 dividing 420). On the other hand, 84 is achievable, by taking
    (e.g.) 1 / 60 + 3 / 70 = 25 / 420 = 5 / 84. So 84 is the answer.