Question

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.​

Answer 1

Step-by-step explanation:

Let a be any. positive integer.

According to Euclid's division lemma

a=3m or 3m+1 or 3m+2

Case I:-

If a =3m, then a²=(3m)²=9m²=3(3m²)=3k, for some positive integer k

Case II:-

If a =3m+1, then a²=(3m+1)²=(3m)²+1²+2(3m)×1

=9m²+1+6m

=9 m²+6m+1

=3(3 m²+2m)+1=3k+1, for some positive integer k

Case III:- If a=3m+2, then a²=(3m+2)²

=(3 m)²+2(3m)×2+2²=9 m²+12m+4

=9 m²+12m+3+1

=3(3 m²+4m+1)+1=3k+1, for some positive integer k

Thus the square of any positive integer is either of the form 3k or 3k+1 for some positive integer k

Answer 2

Answer:

It is the correct answer.

Step-by-step explanation:

Hope this attachment helps you.