Problem 17 · 2019 Math Kangaroo
Stretch
Number Theory
factor-pairssum-constraint
A natural number greater than 0 is written on each side of the die shown (the three visible faces show 10, 15 and 5). All products of opposite numbers are equal. What is the smallest possible sum of all 6 numbers?
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Answer: C — 41
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Hint 1 of 3
Every pair of opposite faces multiplies to the same number; call it k.
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Hint 2 of 3
Then each hidden face is k divided by the face across from it, so k must divide 10, 15 and 5 evenly.
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Hint 3 of 3
Pick the smallest such k to make the hidden faces as small as possible, then add all six.
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Approach: choose smallest common product k
- The three shown faces 10, 15, 5 each have an opposite face, and all three opposite products equal some k.
- Each opposite is k÷10, k÷15, k÷5, so k must be a multiple of 10, 15 and 5 — smallest is k = 30.
- Then the opposites are 3, 2, 6, and the six faces sum to 10+15+5+3+2+6 = 41 (C).
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