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X and y are odd positive integer then prove x square and y square is not divible by 4and also and also X square + Y square is given

Posted by Krushna Khandare 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

Let x = 2p + 1 and y = 2q + 1
{tex}\therefore \quad x ^ { 2 } + y ^ { 2 } = ( 2 p + 1 ) ^ { 2 } + ( 2 q + 1 ) ^ { 2 }{/tex}
{tex}= 4 p ^ { 2 } + 4 p + 1 + 4 q ^ { 2 } + 4 q + 1{/tex}
{tex}= 4 \left( p ^ { 2 } + q ^ { 2 } + p + q \right) + 2{/tex}
{tex}= 2 \left( 2 p ^ { 2 } + 2 q ^ { 2 } + 2 p + 2 q + 1 \right){/tex}
{tex}= 2 m \quad \text { where } m = \left( 2 p ^ { 2 } + 2 q ^ { 2 } + 2 p + 2 q + 1 \right){/tex}
{tex}\therefore \quad x ^ { 2 } + y ^ { 2 }{/tex} is an even number but not divisible by 4.

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