Problem 24 · 2022 Math Kangaroo
Stretch
Number Theory
factorization
Kai puts the numbers 3, 4, 5, 6 and 7 into the five circles of the diagram. For each triangle, the product of the three numbers at its vertices must equal the number written inside that triangle. What is the sum of the numbers at the vertices of the triangle marked 168?
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Answer: D — 17
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Hint 1 of 2
Factor each label as a product of three of the numbers 3,4,5,6,7 to see which three meet in that triangle.
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Hint 2 of 2
The number sitting in every triangle is the shared centre; the 168 triangle uses the other two plus the centre.
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Approach: factor each product to identify the three vertices
- Factor the labels: 105 = 3 × 5 × 7, 84 = 3 × 4 × 7, 210 = 5 × 6 × 7, 168 = 4 × 6 × 7.
- The factor 7 appears in every product, so 7 is the central circle.
- The triangle labelled 168 has vertices 4, 6, 7, summing to 17.
- So the answer is D.
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