If x, y, and z are positive integers such that x is a factor of y, and

Given
We are given 3 positive integers \(x\), \(y\) and \(z\) such that \(x\) is a factor of \(y\) and \(x\) is a multiple of \(z\). We are asked to find that among the options given which of them is not necessarily an integer.

Approach
Since \(x\) is a factor of \(y\), we can say that \(\frac{y}{x}\) is an integer...........(1)

Also, as \(x\) is a multiple of \(z\) i.e. \(z\) is a factor of \(x\), we can say that \(\frac{x}{z}\) is an integer............(2)

From the above two deductions, we can say that \(\frac{y}{z}\) will also be an integer as \(x\) divides \(y\) completely and \(z\) divides \(x\) completely..............(3)

Our endeavor would be to reduce the expressions in the options to one or more of the above 3 forms.

Working Out
(A) \(\frac{x +z}{z} = \frac{x}{z} + 1\) . Since \(\frac{x}{z}\) is an integer, the expression will be an integer

(B) \(\frac{y+z}{x} = \frac{y}{x} + \frac{z}{x}\) . \(\frac{y}{x}\) is an integer but we can't say if \(\frac{z}{x}\) is also an integer. So the expression need not necessarily be an integer.

Although we have got our answer, I am reducing the other expressions for solution purpose.

(C) \(\frac{x+y}{z} = \frac{x}{z} + \frac{y}{z}\). Both \(\frac{x}{z}\) and \(\frac{y}{z}\) are integers, hence the expression will also be an integer

(D) \(\frac{xy}{z}\). Since \(\frac{x}{z}\) is an integer, the expression will also be an integer.

(E) \(\frac{yz}{x}\). Since \(\frac{y}{x}\) is an integer, the expression will also be an integer.

Hence, answer is Option B

Hope this helps

Regards
Harsh