Question

use Euclid division Lemma to show that the square of any positive integer is EITHER OF the of the form 3m or 3m + 1 for some integer m

Answer 1

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Answer 2

Answer:

If a and b are 2 positive integers..

a=bq+r, 0≤r≤b,

let b=3, therefore,r=0,1,2

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

If a=3q ⇒  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+9q+1

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

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