Problem 23 · 2022 Math Kangaroo
Stretch
Number Theory
factorizationcasework
The numbers 1 to 8 are written in the circles shown, one per circle. Along each of the five straight arrows the three numbers in the circles are multiplied, and the product is written at the arrow’s tip. What is the sum of the numbers in the three circles in the bottom row?
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Answer: D — 17
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Hint 1 of 2
Each product label factors into three of the numbers 1-8; the arrow-tip products constrain which numbers sit where.
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Hint 2 of 2
Factor 30, 48, 105, 28, 144 into the digits 1-8 and find the three bottom-row circles in the consistent assignment.
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Approach: factor the arrow products to place numbers
- The numbers 1 to 8 fill the circles (total 36), and each arrow's label is the product of its three circles.
- Factoring a label like \(105 = 3 \times 5 \times 7\) forces those three numbers onto that arrow, and the other labels pin down the rest by elimination.
- The resulting placement puts numbers summing to 17 in the three bottom-row circles, so the answer is D.
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