Question

Use Euclid division Lemma to show that square of any positive numbers of the form 3p for 3 + 1

Answer 1

As you know that

Dividend=Divisor×Quotient+Reaminder

Here take dividend a, Divisor 3, Quotient q and remainder r

a=3q+r

squaring both sides

a^2=9q^2+r^2+6qr

Case 1: Take r=0

Then, a^2=9q^2

a^2=3(3q^2)

a^2=3m where 3q^2=m

Case 2: When r=1

a^2=9q^2+6q+1

a^2=3(3q^2+2q)+1

a^2=3m+1 where m=3q^2+2q

Case 3: When r=2

a^2=9q^2+12q+4

a^2=3(3q^2+4q+1)+1 (by splitting 4 into 3+1)

a^2=3m+1 where m=3q^2+4q+1