Problem 15 · 2009 Math Kangaroo
Hard
Number Theory
last-digitdifference-of-squares
Determine the units digit of the number \(1^2-2^2+\cdots-2008^2+2009^2\).
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Answer: E — 5
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Hint 1 of 2
Only the last digit matters, so reduce each square to its units digit first.
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Hint 2 of 2
Pair the terms as (2009² − 2008²) + … and use a difference-of-squares shortcut on the units digit.
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Approach: work modulo 10 with difference of squares
- Group from the top: (2009² − 2008²) + (2007² − 2006²) + … + (3² − 2²) + 1².
- Each bracket is the sum of two consecutive numbers, and these sums add up so that the units digit settles at 5.
- The unit digit is 5.
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