Question

Check whether the following statement is true or not: 

q : if x, y are integer such that xy is even, then at least one of x and y is an even integer.

Answer 1

Let us Assume that p and q be the statements given by 

p: x and y are integers and xy is an even integer. 

q: At least one of x and y is even. 

Let p be true, then xy is an even integer. So, 

xy = 2(n + 1) 

Now, 

Let x = 2(k + 1) 

Now as x is an even integer, xy = 2(k + 1). y is also an even integer. 

Now take x = 2(k + 1) and y = 2(m + 1) 

xy = 2(k + 1).2(m + 1) = 2.2(k + 1)(m + 1) 

Therefore, it is also true. 

Hence, the statement is true.