1. ## Not An Integer

If x , y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

A. (x + z)/z
B. (y + z)/x
C. (x + y)/z
D. (xy/z)
E. (yz/x)

2. ## Re: Not An Integer

Originally Posted by harpazo
If x , y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

A. (x + z)/z
B. (y + z)/x
C. (x + y)/z
D. (xy/z)
E. (yz/x)

How did you choose?

"x is a factor of y" means what?
"x is a multiple of z" means what?

3. ## Re: Not An Integer

Originally Posted by harpazo
If x , y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

A. (x + z)/z
B. (y + z)/x
C. (x + y)/z
D. (xy/z)
E. (yz/x)

Let $z=1,~x=2,~\&~y=4$ Is B. an integer? Is D.?

4. ## Re: Not An Integer

Saying that "x is a factor or y" means that there exist an integer, p, such that y= px. Saying that "x is a multiple of z" means that there exist an integer, q, such that x= qz. Write everything in terms of z:
A) (x+ z)/z= (qz+ z)/z= (q+ 1)z/z= q+ 1, an integer.
B) (y+ z)/x= (px+ z)/x= p+ z/x. Since x= qz, we can write that as p+ z/(qz)= p+ 1/q which is an integer if and only if q= 1.
C) (x+ y)/z= (qz+ px)/z= (qz+ pqz)/z= (q+ pq)z/z= q+ pq, an integer.
D) (xy)/z= ((qz)y)/z= (qy)z/z= qy, an integer.
E) (yz)/x= ((px)z)/x= pz, an integer.

5. ## Re: Not An Integer

Originally Posted by TKHunny
How did you choose?

"x is a factor of y" means what?

This means y can be divided by x.

"x is a multiple of z" means what?
This means z can be divided by x.

6. ## Re: Not An Integer

Originally Posted by Plato
Let $z=1,~x=2,~\&~y=4$ Is B. an integer? Is D.?
Let z = 1, x = 2, and y = 4.

A. (x + z)/z

(2 + 1)/1 = 3

B. (y + z)/x

(4 + 1)/2 = 5/2

C. (x + y)/z

(2 + 4)/1 = 6

D. (xy/z)

(2•4)/1 = 8

E. (yz/x)

(4•1)/2 = 2

Are they not all integers?