Question

X is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A. x = w
B. x > w
C. x/y is an integer
D. w/z is an integer
E. x/z is an integer

Answer 1

The answer is C.

Explanation:

The key to solve the problem is the equation given

y = 2z which means that y is always an even number.

For example:

let z = 1; y = 2(1) y = 2;

let z = 2; y = 2(2) y = 4;

let z = 3; y = 2(3) y = 6;

And we know that for any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms.

For example:

Set of numbers: [1,2,3,4,5,6] ;

Sum of the set: 1+2+3+4+5+6 = 21 ;

Number of terms of the set: 6;

Therefore, 21 is not a multiple of 6.

But for any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.

For example:

Set of numbers: [1,2,3,4,5,] ;

Sum of the set: 1+2+3+4+5 = 15 ;

Number of terms of the set: 5;

Therefore, 15 is a multiple of 5.

And that's why we know that if y is always an even number (y = 2z), the sum of the consecutive integers (x) is not a multiple of y, therefore x/y cannot be an integer.