Question

Prove that if X&Y are both odd positive integer, then x2+y2 is even but not divisible by 4

Answer 1

Hello Freind
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Review :-
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We know that this all is work by the divisibility rule now let's learn what does it actually mean


Divisibility :-
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A non Zero integer 'a' is said to divide an integer 'a' if there exists an integer c such that b = ac

Now come into our asked question

SOLUTION
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As we know that any odd positive integer like - 3 , 5 .. is of the form of 2q+1 for some integer q in

so let us assume that

x = 2m +1 and y = 2n +1 [ For some integer m and n ]


x² +y² = ( 2m + 1 ) ² + ( 2n + 1 ) ²

= 4 { ( m² +n² ) + ( m + n ) } +2 [ Using formula : ( a+b ) ² = a² +b² + 2ab )


= 4q + 2 , where q = ( m² + n² ) + ( m + n )

Here x² +y² is even and leaves remainder 2 when divided by 4

x² +y² is even but not Divisible by 4

Hence proved
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