Problem 12 · 2017 Math Kangaroo
Hard
Number Theory
Counting & Probability
place-valuecareful-counting
How many positive whole numbers n have the property that exactly one of the two numbers n and n + 20 has four digits?
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Answer: E — 40
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Hint 1 of 2
Four-digit numbers run from 1000 to 9999; when does exactly one of n and n+20 land in that band?
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Hint 2 of 2
Check the two edges of the band separately.
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Approach: find the two boundary windows where exactly one value is four-digit
- At the lower edge: n is three-digit but n+20 is four-digit when n = 980..999 — that is 20 values.
- At the upper edge: n is four-digit but n+20 is five-digit when n = 9980..9999 — another 20 values.
- In between, both are four-digit (not allowed). Total = 20 + 20 = 40, choice E.
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