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)
My answer is D.
Book's answer is B.
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)
My answer is D.
Book's answer is B.
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.