HMMT 二月 2003 · 冲刺赛 · 第 27 题

HMMT February 2003 — Guts Round — Problem 27

[9] 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 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND

解析

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 is relatively prime to 70, it follows that none of the primes 2 , 3 , 7 can divide 7 a + 6 b , so we won’t be able to cancel any of these factors in the denominator. Thus, after reducing 2 to lowest terms, the denominator will still be at least 2 · 3 · 7 = 84 (the product of 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.