Problem 26 · 2023 Math Kangaroo
Stretch
Number Theory
factorizationcasework
Seven pairwise different single-digit numbers are distributed among the circles shown so that the product of the three numbers that are connected by a straight line is the same in all three cases. Which number is written in the circle with the question mark?
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Answer: A — 2
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Hint 1 of 2
The centre number is in all three products, so the two outer numbers on each line have equal products.
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Hint 2 of 2
Find three disjoint pairs of distinct digits with the same product, and see which centre value makes it work.
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Approach: equal products force matching pairs
- All seven digits must be different and non-zero (a zero would force every product to be 0).
- Trying the common product 72 = 8·9·1 = 6·4·3, the two horizontal lines can use {8,9,1} and {6,4,3}.
- The vertical line shares the two centres (9 and 4 in one valid layout), so its third number is 72 ÷ (9·4) = 2.
- Checking all arrangements, the question-mark circle always holds 2.
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