Problem 17 · 2011 Math Kangaroo
Stretch
Number Theory
caseworkplace-value
What is the greatest number of consecutive three-digit numbers that each have at least one odd digit?
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Answer: D — 111
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Hint 1 of 2
A number breaks the run only if all three of its digits are even.
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Hint 2 of 2
Find the longest gap between two consecutive all-even three-digit numbers.
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Approach: find the largest gap between all-even-digit numbers
- A three-digit number fails only when every digit is even (e.g. 688, 800).
- Between 688 and 800 there is no all-even number, since the 7-hundreds all have an odd hundreds digit and the 690–699 block has the odd 9.
- That run is 689 up to 799, which is 799 − 689 + 1 = 111 numbers, each with an odd digit.
- So the longest run is 111, choice (D).
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