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Seeking the Master of Discrete Mathematics to Prove a Problem
1. Proof (direct): Because m, n and p are integers and m+n is even, there are two cases:

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.