If m is an even integer, v is an odd integer, and mv0...

Question

If m is an even integer, v is an odd integer, and m>v>0, which of the following represents the number of even integers less than m and greater than v ?

GMAT Panda · Accepted Answer

m - v - 1/2 We translate “m is an even integer” and “v is an odd integer” into expressions using integer parameters. v = 2j + 1 m = 2k The even integers between v and m start at the first even after v and end at the last even before m. first even = v + 1 = 2(j + 1) last even = m - 2 = 2(k - 1) We count the number of terms in this evenly spaced sequence with step 2. 2(k - 1) - 2(j + 1)/2 + 1 = (k - 1 - (j + 1)) + 1 = k - j - 2 + 1 = k - j - 1 We substitute back k = m/2 and j = v - 1/2 into k - j - 1. k - j - 1 = m/2 - v - 1/2 - 1 We simplify the expression in terms of m and v. = m - (v - 1)/2 - 1 = m - v + 1/2 - 1 = m - v - 1/2

Explanation	Calculations	Help
$We translate “ m is an even integer” and “ v is an odd integer” into expressions using integer parameters.$	$v = 2 j + 1 m = 2 k$
$The even integers between v and m start at the first even after v and end at the last even before m .$	$first even = v + 1 = 2 (j + 1) last even = m - 2 = 2 (k - 1)$
$We count the number of terms in this evenly spaced sequence with step 2.$	$\frac{2 ( k - 1 ) - 2 ( j + 1 )}{2} + 1 = (k - 1 - (j + 1)) + 1 = k - j - 2 + 1 = k - j - 1$
$We substitute back k = \frac{m}{2} and j = \frac{v - 1}{2} into k - j - 1 .$	$k - j - 1 = \frac{m}{2} - \frac{v - 1}{2} - 1$
$We simplify the expression in terms of m and v .$	$= \frac{m - ( v - 1 )}{2} - 1 = \frac{m - v + 1}{2} - 1 = \frac{m - v - 1}{2}$

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