Problem 23 · 2021 Math Kangaroo
Stretch
Number Theory
digit-sumplace-value
Let N be the smallest positive integer whose digits sum to 2021. What is the sum of the digits of \(N+2021\)?
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Answer: A — 10
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Hint 1 of 2
To keep N smallest with digit sum 2021, use as many 9s as possible with a small leading digit.
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Hint 2 of 2
Adding 2021 to a string of trailing 9s triggers carries that turn many 9s into 0s.
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Approach: build the smallest N, then add and track carries
- 2021 = 224×9 + 5, so the smallest N is a 5 followed by 224 nines (digit sum 2021).
- Adding 2021 to the trailing ...9999 causes carries that ripple through every 9, turning them to 0 and bumping the leading 5 to 6.
- N+2021 becomes 6, then a block of zeros, ending in 2020.
- Its digit sum is 6+2+0+2+0 = 10, choice (A).
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