Difference between revisions of "1986 AJHSME Problems/Problem 17"
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==Solution== | ==Solution== | ||
− | + | We can solve this problem using logic. | |
+ | |||
+ | Let's say that <math>n</math> is odd. If <math>n</math> is odd, then obviously <math>no</math> will be odd as well, since <math>o</math> is odd, and odd times odd is odd. Since <math>o</math> is odd, <math>o^2</math> will also be odd, because <math>o^2 = oo</math>, and odd times odd is odd. And adding two odd numbers makes an even number, so if <math>n</math> is odd, the entire expression is even. | ||
+ | |||
+ | Let's say that <math>n</math> is even. If <math>n</math> is even, then <math>no</math> will be even as well, because odd times even is even. <math>o^2</math> will still be odd. That means that the entire expression will be odd, since odd + even = odd. | ||
+ | |||
+ | Looking at the multiple choices, we see that our second case fits choice E exactly. | ||
+ | |||
+ | E | ||
==See Also== | ==See Also== | ||
[[1986 AJHSME Problems]] | [[1986 AJHSME Problems]] |
Revision as of 19:24, 24 January 2009
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
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 odd times odd is odd. Since
is odd,
will also be odd, because
, and odd times odd is 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 odd times even is even.
will still be odd. That means that the entire expression will be odd, since odd + even = odd.
Looking at the multiple choices, we see that our second case fits choice E exactly.
E