Question

Use Euclid's division lemma to show that they possess of
the form 3m or 3m + 1 for some integer m.​

Answer 1

Let a be any possitive integer and b=3.

By eucluds division lemma,

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

Case I- a=3c

a²=9c²=3(3c²)=3m

Case II-a=3c+1

a²=9c²+6c+1=3(3c²+2c)+1=3m+1

Case III-a=3c+2

a²=9c²+6c+4=9c²+6c+3+1=3(3c²+2c+1)+1=3m+1

Therefore square of any possitive integer is of the form 3m or 3m+1.

Now let me help you in judging why I started with a and not directly with a².This is because of the language of question.The question says square of a possitive integer which implies that first we have to take the possitive integer and then square it.

To check your understanding,try to prove

The cube of a possitive Integer is of the form of 9m,9m+1,9m+8.(HINT-Read language of question)

Cheers