2013 UNCO Math Contest II Problems/Problem 5

Problem

If the sum of distinct positive integers is $17$ , find the largest possible value of their product. Give both a set of positive integers and their product. Remember to consider only sums of distinct numbers, and not $3+7+7$ or $2+3+4+4+4$ , etc., which have repeated terms. You need not justify your answer on this question.

$\begin{array}{|c|c|c|c|} \hline \text{EXAMPLE: }& \text{Distinct Integers: }{2, 3, 4, 8} & \text{Their Sum: }2+3+4+8=17 & \text{Their Product: }2 \times 3\times 4\times 8=192 \\ \hline \end{array}$

Solution

Intuitively, we want all the numbers to be as close as possible to $\tfrac{17}4$ , or $4.5$ . Thus, we get the numbers $2$ , $4$ , $5$ , and $6$ . These multiply up to $2\cdot4\cdot5\cdot6 = \boxed{240}$ .

~pineconee

2013 UNCO Math Contest II (Problems • Answer Key • Resources)
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2013 UNCO Math Contest II Problems/Problem 5

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