number theory

Prove that the number nm(n-m) is even for any integers n & m

4 Answers

If n , m odd
the deffirence is even,,,,,

odd*even=even

Case 1,
If n and m are both even.
nm is even and n-m is even
So, nm(n-m) is also even.

Case2,
If 1 is odd and the other is even
nm is even. n-m is odd
so, nm(n-m) is even.

Case3,
If both are odd.
nm is odd. n-m is even
again nm(n-m) is even.

Hence proved. :)

341

Hari Shankar ·2011-08-04 02:25:21

The sum of the three given numbers = n + m + (n-m) = 2n is even.

Hence all three cannot be odd.

Hence their product is even

that was much simpler thanks

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