1986 AJHSME Problems/Problem 17
Let be an odd whole number and let be any whole number. Which of the following statements about the whole number is always true?
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.
We are given that , so in mod we have which is odd only if is even
Solution 3 (easiest)
To make this problem simpler, we can assume a number to replace and . Let be and be . When we compute , we get . We immediately rule out , , and . The only options left are and . This time let's assume is , and is . comes out to be . isn't odd, so we cross out . Thus, the answer is
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