Problem 19 · 2015 Math Kangaroo
Hard
Number Theory
digit-sum
There are 2015 marbles in a pipe. They are numbered 1 to 2015. Marbles whose digits add up to the same number have the same colour, and marbles whose digits have a different sum have a different colour. How many different colours do the marbles in the pipe have?
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Answer: C — 28
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Hint 1 of 2
The colour of a marble depends only on its digit sum.
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Hint 2 of 2
Find the smallest and largest digit sum occurring among 1 to 2015 — then count the values in between.
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Approach: count the distinct digit sums from 1 to 2015
- Colours correspond exactly to the different digit sums that appear.
- The smallest digit sum is 1 (e.g. 1, 1000) and the largest is 28 (from 1999).
- Every value from 1 to 28 is achievable, so there are 28 colours (C).
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