Problem 27 · 2013 Math Kangaroo
Stretch
Number Theory
arithmetic-series
If you add the first n positive integers, you get a three-digit number with all three digits the same. What is the digit sum of n?
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Answer: B — 9
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Hint 1 of 2
The sum of the first n integers is n(n+1)/2; the target is a repdigit like 111, 222, …
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Hint 2 of 2
Test which of 111–999 is triangular.
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Approach: triangular number that is a repdigit
- n(n+1)/2 must be one of 111,222,…,999.
- 666 works: n(n+1)=1332 gives n=36.
- Digit sum of 36 is 3+6 = 9, so B.
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