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Ede4ka [16]
3 years ago
13

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

Mathematics
1 answer:
Ray Of Light [21]3 years ago
6 0

Step-by-step explanation:

The proof can be done by contradiction. Suppose both a, and b weren't even. So that a, and b are both odd. This means they both look like

a=2k+1,~~b=2l+1 (for some integers k and l)

So, let's compute what a^2-3b^2 would be in this case:

a^2-3b^2=(2k+1)^2-3(2l+1)^2=4k^2+4k+1-3(4l^2+4l+1)

= 4k^2+4k+1-12l^2-12l-3=4k^2+4k-12l^2-12l-2

=4(k^2+k+3l^2-3l)-2

which notice wouldn't be divisible by 4. This shows then that since a^2-3b^2 is divisible by 4, at least one of the integers a and b is even.

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Greetings !

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