Bunuel wrote:

Official Solution:

If \(x\) and \(y\) are positive integer and \(xy\) is divisible by 4, which of the following must be true?

A. If \(x\) is even then \(y\) is odd.

B. If \(x\) is odd then \(y\) is a multiple of 4.

C. If \(x+y\) is odd then \(\frac{y}{x}\) is not an integer.

D. If \(x+y\) is even then \(\frac{x}{y}\) is an integer.

E. \(x^y\) is even.

Notice that the question asks which of the following MUST be true, not COULD be true.

A. If \(x\) is even then \(y\) is odd. Not necessarily true, consider: \(x=y=2=\text{even}\);

B. If \(x\) is odd then \(y\) is a multiple of 4. Always true: if \(x=\text{odd}\) then in order \(xy\) to be a multiple of 4 \(y\) must be a multiple of 4;

C. If \(x+y\) is odd then \(\frac{y}{x}\) is not an integer. Not necessarily true, consider: \(x=1\) and \(y=4\);

D. If \(x+y\) is even then \(\frac{x}{y}\) is an integer. Not necessarily true, consider: \(x=2\) and \(y=4\);

E. \(x^y\) is even. Not necessarily true, consider: \(x=1\) and \(y=4\);

Answer: B

Hi Buneul,

As per the question, X and Y are positive number and XY is divisible by 4. Then we have to consider numbers which are only divisible by 4.

Then how Choice (B) can be correct.