Problem 29 · 2020 Math Kangaroo
Stretch
Number Theory
Algebra & Patterns
factorizationsubstitution
Sonia writes three consecutive whole numbers, one on each side of a triangle. Then, on each vertex of the triangle, she writes the sum of the numbers written on the two sides that meet at that vertex, and she multiplies these three vertex numbers, obtaining the product 504. What is the product of the three numbers written on the sides of the triangle?
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Answer: B — 60
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Hint 1 of 2
Call the three consecutive numbers n, n+1, n+2; each vertex product is a sum of two of them.
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Hint 2 of 2
The vertex sums are 2n+1, 2n+2, 2n+3; set their product to 504 and find n.
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Approach: set up the vertex products
- With sides n, n+1, n+2, the three vertices hold (2n+1), (2n+2), (2n+3), and their product is 504.
- Testing, 7 × 8 × 9 = 504, so 2n+1 = 7, giving n = 3 and sides 3, 4, 5.
- The product of the side numbers is 3 × 4 × 5 = 60.
- The answer is 60, choice B.
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