2010/11 AHSMC Part 1 Problem 12

Problem 12 of 16

A gold number is a positive integer which can be expressed in the form $ab + a + b$ , where $a$ and $b$ are positive integers. The number of gold numbers between $1$ and $20$ inclusive is

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Note that $ab + a + b + 1 = (a + 1)(b + 1)$ . Every composite number can be written in this form and no prime number can be written in this form. Therefore, the positive integers that are not gold numbers are those that are one less than a prime. By simple counting, we see there are $8$ primes from $2$ to $21$ . Therefore, the number of gold numbers between $1$ and $20$ inclusive is $20 - 8 = 12$ . The answer is (E).

University of Alberta

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