2011/12 AHSMC Part 1 Problem 9

Problem 9 of 16

A lucky number is a positive integer $n$ such that $7$ is the largest divisor of $n$ which is less than $n$ . The number of lucky numbers is

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Solution

Since a lucky number $n$ is divisible by $7$ , it has the form $n = 7k$ for some positive integer $k$ . If $k$ is not a prime number, then it has a divisor $h$ where $1 < h < k$ , and $7h$ is a divisor of $n$ larger than $7$ but not equal to $n$ . Hence $k$ must be a prime number. Moreover, it cannot be greater than $7$ . Hence there are only $4$ lucky numbers, namely, $14$ , $21$ , $35$ and $49$ . The answer is (D).

University of Alberta

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