Problem 17 · 2025 Math Kangaroo
Hard
Number Theory
divisibilitydigit-sum
The five-digit number \(\overline{N18NN}\) is divisible by 18. Which of the following statements is true for the digit N?
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Answer: A — There is exactly one such N.
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Hint 1 of 2
Use the rules for divisibility by 2 and by 9 separately.
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Hint 2 of 2
Last digit N must be even, and the digit sum 3N + 9 must be a multiple of 9, forcing N to be a multiple of 6.
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Approach: split 18 = 2×9 into two digit tests
- Divisible by 2: N is even. Divisible by 9: N + 1 + 8 + N + N = 3N + 9 is a multiple of 9, so N is a multiple of 3.
- Even and multiple of 3 means N is a multiple of 6; as a nonzero leading digit, only N = 6 works.
- So there is exactly one such N.
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