Question

If x and y are odd Integers, then xy is odd

Answer 1

Yes, this statement is true. If x and y are both odd integers, then both x and y are of the form 2n + 1 for some integer n. The product of two odd numbers is still odd because (2n + 1)(2m + 1) = 4nm + 2(n + m) + 1, which is an odd integer.

Answer 2

Answer: True

Some examples would be

3*5 = 15

7*9 = 63

11*11 = 121

The template is odd*odd = odd.

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Here is the proof as to why this is the case.

x = 2k+1 where k is some integer

y = 2m+1 where m is some integer

Both x and y are odd since we added 1 to an even number.

Multiply out x and y

xy = (2k+1)*(2m+1)

xy = 2k(2m+1) + 1*(2m+1)

xy = 4km+2k + 2m+1

xy = 2*(2km+k+m) + 1

xy = 2n + 1 where n = 2km+k+m is an integer

The result is of the form 2*(integer)+1 which is odd. This concludes the proof.