Q. 115.0( 2 Votes )

Prove that

Answer :

We know that any odd positive integer is of the form


2p + 1;


where p = 0,1,2,…….


Therefore Let x = 2m + 1 and y = 2n + 1 for any integers m and n


Now, x2 – y2 = (2 m + 1)2 – (2 n + 1)2


x2 – y2 = 4 m2 + 4 m + 1 – 4 n2 – 4 n – 1


x2 – y2 = 4 m2 + 4 m – 4 n2 – 4 n


x2 – y2 = 4 (m2 – n2 + m – n)


x2 – y2 = 4 (m – n)(m + n + 1)


x2 – y2 = 4p; where p = (m – n)(m + n + 1)


Now when we divide x2 – y2 by 4 leaves no remainder


Hence, x2 – y2 is divisible by 4.


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