Problem 18 · 2016 Math Kangaroo
Hard
Number Theory
digit-sumdivisibility
Three three-digit numbers are built using the digits 1 to 9 so that each of the nine digits is used exactly once. Which of the following numbers cannot be the sum of the three numbers?
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Answer: A — 1500
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Hint 1 of 2
The sum of all nine digits is fixed.
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Hint 2 of 2
1+2+...+9 = 45, so the total of the three numbers must be a multiple of 9.
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Approach: digit-sum divisibility by 9
- The nine digits 1..9 sum to 45, a multiple of 9, so the total of the three numbers is divisible by 9.
- Among the options only 1500 is not a multiple of 9 (1+5 = 6), so it cannot be the sum (A).
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