Prove that when we divide the square of a natural number by 8, the remainder obtained will be 0 or 1 or 4
Proof:
Let x be a natural number
Case 1
x= even number
x2 =(2n)2
=4n2
ie,
x2/8=4n2/8
= n2/2
=Quotient+0/2 or Quotient+1/2
= Quotient+0/8 or Quotient+4/8
There fore remainder = 0 or 4
when we divide ,a square of an even number by 8, the remainder is either 0 or 4
Case 2
x=odd number
x2=(2n+1)2
=4n2+4n+1
=4n(n+1)+1
ie,
x2/8=(4n(n+1)+1)/8
=n(n+1)/2 + 1/8
=natural number +1/8
Therefore remainder =1
when we divide ,a square of an odd number by 8, the remainder is 1
Hence ,when we divide ,a square of a natural number by 8, the remainder is 0 or 1 or 4
