Prove that for all integers m and n, m + n and m - n are either both odd or both even.?
回答 (1)
2015-11-15 8:44 pm
take numbers mod 2
even ≡ 0 mod 2
odd ≡ 1 mod 2
m ≡ p mod 2
n ≡ q mod 2
m+m ≡ p+q mod 2
m-n ≡ p-q mod 2
if p≡q then p-q is 0
p+q = 0 or 2. Both are ≡ 0 mod 2.
if p not ≡ q mod 2 then p+q ≡ 1 mod 2
p-q ≡ 1 or -1. Both are ≡ 1 mod 2.
Therefore both are even or both are odd.
QED
even ≡ 0 mod 2
odd ≡ 1 mod 2
m ≡ p mod 2
n ≡ q mod 2
m+m ≡ p+q mod 2
m-n ≡ p-q mod 2
if p≡q then p-q is 0
p+q = 0 or 2. Both are ≡ 0 mod 2.
if p not ≡ q mod 2 then p+q ≡ 1 mod 2
p-q ≡ 1 or -1. Both are ≡ 1 mod 2.
Therefore both are even or both are odd.
QED
收錄日期: 2021-04-21 15:09:56
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20151115120100AAuSIve