1.m and n are even numbers. Since n+p is even and p is even, m+p is even.
The sum of two even numbers must be an even number. )
M and n are both odd numbers. Since n+p is even and p is odd, m+p is even.
The sum of two odd numbers must be odd. )
To sum up, it proves that.
2. Proof (reduction to absurdity): If m+p is not odd, then one of M and P must be even and the other is odd.
If m is even and p is odd, because m+n is even, so n must be even, then n+p is odd.
A Number that is inconsistent with n+p being an even number is not valid.
If p is an even number and m is an odd number, it can also prove that the hypothesis is not true.
To sum up, it proves that.