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Prove that if x and y are both odd positive integers then x2+y2 is even but not divisible by 4

Posted by Ayush Nagar 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

Since x and y are odd positive integers, Then it should be in the form of 2x+1 (where x is a positive integer)

Let x = 2m + 1 and y = 2n + 1 ( where m and n positive integers)

Now x² + y²

= (2m + 1)² + (2n + 1)²

= 4m² + 4m + 1 + 4n² + 4n + 1

= 4(m² + n² + m + n) + 2 ....... (1)

from (1) we get

x² + y² = 2 {m² + n² + m + n) + 1} = 2t (t = 2(m² + n² + m + n) + 1 is a positive integer

Therefore it is clear that {tex}x^2+y^2{/tex} is an even number but not divisible by 4.

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CBSE > Class 10 > Mathematics