Problem 4 · 2023 Math Kangaroo
Medium
Number Theory
place-valuedigit-sum
Let A be a 2023-digit number where every digit is 1. What is the sum of the digits of the number \(A \cdot 1111\)?
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Answer: D — 8092
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Hint 1 of 2
Multiplying by 1111 is the same as adding the number to itself shifted by 1, 2 and 3 places.
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Hint 2 of 2
Watch how the overlapping 1's add up (and carry) in the middle versus at the two ends.
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Approach: model A·1111 as four shifted copies added, then sum the digits
- A·1111 = A·1000 + A·100 + A·10 + A, i.e. four copies of the repunit shifted by 0,1,2,3.
- The middle columns each receive four 1's, producing repeating 4's with regular carries; only the ends differ.
- Carrying everything out, the digit sum of A·1111 works out to 8092.
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