Question

Use Euclid's division lemma to show that the square of any positive integer is of the form 3p, 3p+1.​

Answer 1

by using euclid's division algorithm

a=bq+r a can be any positive integer and b is equal to 3

a= 3q+r r= 0,1,2.

a=3q, 3q+1, 3q+2

if a=3q = (3q)^2

= 9q^2

= 3(3q^2)

= 3p where p= 3q^2

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

= 9q^2 + 6q+ 1

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

= 3p+1 where p= 3q^2 +2q.

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

= 9q^2 +12q +4

= 9q^2 +12q +3+1

= 3(3q^2 +4q )+1

= 3p+1 where p= 3q^2 +4q

this shows that the square of any positive integer is of the form 3p and 3p+1.

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