2015 UNCO Math Contest II Problems/Problem 6

Problem

How many ordered pairs $(n,m)$ of positive integers satisfying $m < n \le 50$ have the property that their product $mn$ is less than $2015$ ?

Solution

We can check every possible case by considering m = 1 and n = 2 through 50, m = 2 and n = 3 through 50, etc. Note that $40(50)=2000$ whereby it is obvious that for m = 1 through 40, all possible n work. This accounts for $49+48\dots+10=\frac{49(50)}{2}-45=1180$ cases. We individually check the remaining cases. Note that $41(49)=2009$ and $41(50)=2050$ so m = 41 contributes $49-42+1=8$ cases. Similarly, m = 42 contributes 5 cases, m = 43 contributes 3 cases, and m = 44 contributes 1 case. This sums to $\boxed{1197}$ .

2015 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2015 UNCO Math Contest II Problems/Problem 6

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