If x < y < z and y-x > 9, where x is an even integer and y and z are odd integers, what is the least possible value of z - x?  Choose the correct choice from the following options:
A) 6  B) 7  C) 8  D) 13  E) 10

Rationale: we have : 1 ) x < y < z 2 ) y - x > 9 3 ) x = 2 k ( x is an even number ) 4 ) y = 2 n + 1 ( y is an odd number ) 5 ) z = 2 p + 1 ( z is an odd number ) 6 ) z - x = ? least value z - x = 2 p + 1 - 2 k = 2 p - 2 k + 1 = 2 ( p - k ) + 1 - that means that z - x must be an odd number . we can eliminate answer choices a , c and e we are asked to find the least value , so we have to pick the least numbers since y is odd and x is even , y - x must be odd . since y - x > 9 , the least value for y - x must be 11 , the least value for x must be 2 , and , thus , the least possible value for y must be 13 ( y - 2 = 11 , y = 13 ) 2 < 13 < z , since z is odd , the least possible value for z is 15 z - x = 15 - 2 = 13 answer d
Answer: D