29.2k views
2 votes
If 4 divides a^2-3b^2, then at least one of the integers a and b is even.

User R J
by
7.8k points

1 Answer

2 votes

Answer:

Explanation:

Suppose, on the contrary, a and b are both odd integers, that is:


a=2n+1\\\\b=2m+1\\\\

m and n being some integers numbers.

This way you have to:


a^(2)-3b^(2)=(2n+1)^2-3(2m+1)^2=(4n^2+4n+1)-3(4m^2+4m+1)\\\\a^(2)-3b^(2)=4(n^2-3m^2+n-3m)-2=4(n(n+1)-3m(m+1))-2

The last expression cannot be divisible by 4 since 2 is not divisible by 4. The previous conclusion leads to a contradiction, which was generated from the assumption that a and b were both odd integers. In conclusion, at least one of the two a and b should be an even integer

User Rtenha
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories