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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.

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We have given two integers x and y and it is also given that both x and y are positive odd integers and we have been asked to prove that x²+y² is even number but it is not divisible by 4.

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 1A, question no 9


1 Answer

  1. We know that any odd positive integer is of the form 2q+1, where q is an integer.

    So, let x=2m+1 and y=2n+1, for some integers m and n.

    we have +




    +=4q+2, when q=(+)+(m+n)

    + is even and leaves remainder 2 when divided by 4.

    + is even but not divisible by 4.

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