Proof : when we divide the square of a natural number by 8, the remainder obtained will be 0 or 1 or 4

We don’t need the help of Lagrange’s theorem to 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