Question

show that square of any positive integer can be represent in the form of 3q + 1 or 3q + 2​

Answer 1

Answer:

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class 10

maths

real numbers

a positive integer is the form of 3q+1 q, being a natural number. can you write its square in any form other than 3m+1 i.e. 3m or 3m+2 for some integer? justify your answer.

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text solution

solution :

no, by euclid lemma, b=aq+rm

<br> here, b is any positive integer

0 le r lt 3

3q,3q+1 or 3q+2

(3q)^(2) =9q^(2)=3m

m=3q^(2)

(3q+1)^(2)=9q^(2)+6q+1

=3(3q^(2)+2q)+1=3m+1

m=3q^(2)+2q

(3q+2)^(2)=9q^(2)+12q+4

=9q^(2)+12q+3+1

3=(3q^(2)+4q+1)

3m=1

m=3q^(2)+4q+1]