M08-24

Official Solution:

If \(x\) is a positive odd integer, which of the following numbers must be even?

A. \(x^{\frac{6}{3} }\)


B. \(\frac{x^2 - 1}{2}\)


C. \(xp + 1\), where \(p\) is a prime number


D. \(\frac{(x + 7)(x - 2)}{2}\)


E. \(x^{x - 1}\)


Let's evaluate each option:

A. \(x^{\frac{6}{3} }\). This is equivalent to \(x^2\), which will be odd because the square of an odd number is odd (\(odd^2 = odd\)).

B. \(\frac{x^2 - 1}{2}\). This is equivalent to \(\frac{x^2 - 1}{2} = \frac{(x - 1)(x + 1)}{2}\). Since \(x\) is odd, both \((x - 1)\) and \((x + 1)\) are even, so their product is a multiple of 4. Dividing by 2 gives an even result.

C. \(xp + 1\), where \(p\) is a prime number. If \(p = 2\), then \(xp + 1 = odd*even + 1 = odd\).

D. \(\frac{(x + 7)(x - 2)}{2}\). Since \(x\) is odd, the expression becomes \(\frac{even*odd}{2}\). If that even number, \((x + 7)\), isn't a multiple of 4, the overall expression will be odd. For example, consider \(x = 3\).

E. \(x^{x - 1}\). Given that \(x\) is a positive odd number, the expression equals to \(\text{positive odd number}^even\), which is odd.


Answer: B