clipea12
If x and y are positive integers, what is the remainder when xy is divided by 4?
(1) x + y = xy
(2) (x – 1)(y – 1) is odd.
Here, it's best to test cases for each statement.
In statement (1), we know that x+y = xy. If x and y are both odd, then x+y is even yet x*y is odd. So this case is not possible.
If x+y are both even, then x+y is even as is x*y.
Lastly, if x and y have different signs (order does not matter), then x+y is odd while x*y is even. The two sides of the equal sign do not match, so this case is not possible.
As a result, the only case that works occurs when both x and y are even. If x and y are both even, the their product is divisible by four since the product of two even numbers has at least two 2's in its prime factors.
Sufficient.
In statement (2), odd*odd = odd, so x and y are each one greater than an odd number. This fact means that x and y are even. The product of two even number is even. Sufficient.