Difference between revisions of "1986 AJHSME Problems/Problem 17"
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[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
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Revision as of 20:23, 3 July 2013
Contents
[hide]Problem
Let be an odd whole number and let be any whole number. Which of the following statements about the whole number is always true?
Solution
Solution 1
We can solve this problem using logic.
Let's say that is odd. If is odd, then obviously will be odd as well, since is odd, and the product of two odd numbers is odd. Since is odd, will also be odd. And adding two odd numbers makes an even number, so if is odd, the entire expression is even.
Let's say that is even. If is even, then will be even as well, because the product of an odd and an even is even. will still be odd. That means that the entire expression will be odd, since the sum of an odd and an even is odd.
Looking at the multiple choices, we see that our second case fits choice E exactly.
Solution 2
We are given that , so in mod we have which is odd only if is even
See Also
1986 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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