Question

A positive integer is pf the form 3q+1 , q being a natural number can you write its square in any form other than 3m+1 that is 3m or 3m+2 for some integer m? Justify your answer

Answer 1

Euclid division Lema states that
$a = bq + r$
according to Euclid Lema
a=3q+1
given that q is an natural number
so q is equal to 1,2,3,4.....
so square it Will be of the form
3m+4,3m+6....

Answer 2

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Note: the numbers after variables are their powers.

It is necessary to solve all the values of r in the exam

Answer:

By using Euclid's Division Lemma, a=bq+r

Where, 0 ≤ r < b here, b=3 therefore, r= 0,1 or 2

So,

1. r= 0

2. r= 1

(skipping to r= 2 NOT TO BE DONE IN EXAM)

3. r= 2

a²= (3q+2)²

a²= 9q² + 12q + 4

a²= 3(3q² + 4q) + 4

Now, let (3q2 + 4q) be m

Therefore, a²= 3m + 4.