HMMT 二月 2020 · 冲刺赛 · 第 7 题

HMMT February 2020 — Guts Round — Problem 7

[5] Ana and Banana are rolling a standard six-sided die. Ana rolls the die twice, obtaining a and a , 1 2 then Banana rolls the die twice, obtaining b and b . After Ana’s two rolls but before Banana’s two rolls, 1 2 they compute the probability p that a b + a b will be a multiple of 6. What is the probability that 1 1 2 2 1 p = ? 6

解析

[5] Ana and Banana are rolling a standard six-sided die. Ana rolls the die twice, obtaining a and a , 1 2 then Banana rolls the die twice, obtaining b and b . After Ana’s two rolls but before Banana’s two 1 2 rolls, they compute the probability p that a b + a b will be a multiple of 6. What is the probability 1 1 2 2 1 that p = ? 6 Proposed by: James Lin 2 Answer: 3 1 Solution: If either a or a is relatively prime to 6, then p = . If one of them is a multiple of 2 1 2 6 1 1 but not 6, while the other is a multiple of 3 but not 6, we also have p = . In other words, p = 6 6 2 2 (3 − 1)(2 − 1) 1 1 2 if gcd( a , a ) is coprime to 6, and otherwise p 6 = . The probability that p = is = 1 2 2 6 6 6 3 2 q − 1 where corresponds to the probability that at least one of a and a is not divisible by q for q = 2, 3. 2 1 2 q