HMMT 二月 2008 · 冲刺赛 · 第 23 题

HMMT February 2008 — Guts Round — Problem 23

[ 10 ] Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer n < 60 (both Kelly and Jason know that n < 60, but that they don’t know what the value of n is). The computer tells Kelly the unit digit of n , and it tells Jason the number of divisors of n . Then, Kelly and Jason have the following dialogue: Kelly: I don’t know what n is, and I’m sure that you don’t know either. However, I know that n is divisible by at least two different primes. Jason: Oh, then I know what the value of n is. Kelly: Now I also know what n is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of n ?

解析

[ 10 ] Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer n < 60 (both Kelly and Jason know that n < 60, but that they don’t know what the value of n is). The computer tells Kelly the unit digit of n , and it tells Jason the number of divisors of n . Then, Kelly and Jason have the following dialogue: Kelly: I don’t know what n is, and I’m sure that you don’t know either. However, I know that n is divisible by at least two different primes. Jason: Oh, then I know what the value of n is. Kelly: Now I also know what n is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of n ? Answer: 10 The only way in which Kelly can know that n is divisible by at least two different primes is if she is given 0 as the unit digit of n , since if she received anything else, then there is some number with that unit digit and not divisible by two primes (i.e., 1 , 2 , 3 , 4 , 5 , 16 , 7 , 8 , 9). Then, after Kelly says the first line, Jason too knows that n is divisible by 10. The number of divisors of 10 , 20 , 30 , 40 , 50 are 4 , 6 , 8 , 8 , 6, respectively. So unless Jason received 4, he cannot otherwise be certain of what n is. It follows that Jason received 4, and thus n = 10.