Question

Use Euclids division lemma to show the square of any positive integers is either of the form 3n or 3n+1 for some integer n.

Answer 1

Here your answer goes

Euclid Division Lemma :-

• It is basically a theorem given be Euclid
• In this  Division lemma Divisior = Dividend * Quotient + Remainder
• a = bq + r
• Where , 0 $\leq$ r ∠ b

====>>>> The Proof of your Question is attached with my answer

Refer to the attachments with my answer

NOTE :- I Considered integer as m instead of  " n "

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Be Brainly

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Answer 2

let us take, x= 3m , 3m+1, 3m+2

when, x=3m

        x^2 =  (3q)^ 2

         x^2 = 9q^2 

        x^2  = 3(3q^2)

we see that 3q^2= n

so we have done the first equation 3n

when x=3m+1

           x^2= (3q+1)^2

           [(a+b)^2 = a^2+2ab+b^2]

           x^2= 9q+6q+1

           x^2= 3(3q+2q)+1

 3q+2q= n

    this satisfy the equation m+1

@skb