Question

Prove that square of any positive integer is a form of 3m or 3m+1 3m+2

Answer 1

Step-by-step explanation:

To prove:-The square of any positive integers is in the form 3m or 3m+1.

Proof:-

Using Euclid division Lemma.

If we divide any integer x by 3,it gives a quotient q and remainder r.

Where,

x=3q+r 0≤r<3

So x=3q or x=3q+1 or x=3q+2

Case 1:-If x=3q

Square of x is

x²=(3q)²

x²=9q²

x²=3(3q²)

Let 3q²=m

x²=3m

It's in the form of 3m

Case 2:-If x=3q+1

Square of x is

x²=(3q+1)²

x²=9q²+6q+1

x²=3(3q²+2q)+1

Let 3q²+2q be m

x²=3m+1

It's in the form of 3m+1.

Case 3:-If x=3q+2

Square of x is

x²=(3q+2)²

x²=9q²+12q+4

x²=9q²+12q+3+1

x²=3(3q²+4q+1)+1

Let 3q²+4q+1 be m

x²=3m+1

It's in the form of 3m+1.