Using Euclid's division lemma, show that the square of any positive integer is either of the form 3 m or 3m +1 for some integer m

ANSWER PLEASE!

Sneha

Exploring new things...

Saturday, 28 November 2020 08:38 AM

Let 'a' be any positive integer.

On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.

Such that ,

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

When, r = 0

? a = 3q

When, r = 1

? a = 3q + 1

When, r = 2

? a = 3q + 2

When , a = 3q

On squaring both the sides,

When, a = 3q + 1

On squaring both the sides ,

When, a = 3q + 2

On squaring both the sides,

Therefore , the square of any positive integer is either of the form 3m or 3m+1.

On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.

Such that ,

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

When, r = 0

? a = 3q

When, r = 1

? a = 3q + 1

When, r = 2

? a = 3q + 2

When , a = 3q

On squaring both the sides,

When, a = 3q + 1

On squaring both the sides ,

When, a = 3q + 2

On squaring both the sides,

Therefore , the square of any positive integer is either of the form 3m or 3m+1.