CEMC Cayley

Algebra Difficulty 2.4 Junior Find the answer

A line has equation $y=mx-50$ for some positive integer $m$ . The line passes through the point $(a, 0)$ for some positive integer $a$ . What is the sum of all possible values of $m$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since the line with equation $y=mx-50$ passes through the point $(a, 0)$ , then $0=ma-50$ or $ma=50$ . Since $m$ and $a$ are positive integers whose product is 50, then $m$ and $a$ are divisor pair of 50. Therefore, the possible values of $m$ are the positive divisors of 50, which are $1,2,5,10,25,50$ . The sum of the possible values of $m$ is thus $1+2+5+10+25+50=93$ .

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