Question

Use euclids division algorethem to show that square of every positive intezer is either of the form 3m or 3m+1

Answer 1

Heyaa♡♡

Here it is:-

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,

a2 = 9q2

= 3(3q2)

= 3m , where m = 3q2

If a = 3q + 1

a2 = 9q2 + 6q + 1

= 3(3q2 + 2q) + 1

= 3m + 1 , where m = 3q2 + 2q

If a = 3q + 2 a2

= 9q2 + 12q + 4

= 3(3q2 + 4q + 1) + 1

= 3m + 1, where m = 3q2 + 4q + 1

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

Hope it helped you.