Question

A positive integer is of the form 3q+1,qbeing a natural number.can you write it's square in any form other than 3m+1,that is,3m or 3m+2for some integer m .justify your answer

Answer 1

using Euclid division lemma a=bq +r
a=3q+ r
r=0,1,2
3q +1 square
9q square +1 + 2*3*1
9q square +1+6
take comman from it
it came 3 [3q+2] +1 where [3q+2=m]
:. it came 3q+1
similarly prove second one also ok
then wright statement down side ok hope it will help you if you like then mark me brain list please ok have a nice day

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 (AnsWer)