Problem 19 · 2010 Math Kangaroo
Stretch
Number Theory
factor-pairs
The side lengths of a triangle in cm are given by the natural numbers 13, x and y. Determine the perimeter of the triangle if xy = 105.
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Answer: A — 35
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Hint 1 of 2
List the ways to factor 105 into x times y.
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Hint 2 of 2
Then keep only the pair that forms a valid triangle with side 13.
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Approach: factor 105, then apply the triangle inequality
- Factor pairs of 105: (1,105), (3,35), (5,21), (7,15).
- The sides must satisfy the triangle inequality with 13; only 7, 15, 13 works (8 < 13 < 22).
- The perimeter is 13 + 7 + 15 = 35.
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