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?
i did as:
all options except B are integer so B is the answer. Is there any other good approach?
x is factor of y so \(y = x*A\) where A is another positive integer
x is a multiple of z, so \(x = B*z\) where B is another positive integer and \(y=A*B*z\)
Now \((x+z)/z = x/z + 1 = B+1\) so integer
\((y+z)/x = (A*B*z+z)/(B*z) = A + 1/B\) which will not be an integer if B is an integer greater than 1, so Answer B
We can quickly see that C reduces to \(B + A*B\), so integer, D is \(A*B^2\), so integer and E is \(A*z\), so integer again.
It looks elaborate but can be done in less than 90 seconds with pen and paper and would always give right answer, whereas trying to plug numbers may or may not work.