Question

Use Euclid's division lemma to show
that the square of any positive integer
is of the form 3p, 3p + 1.
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Answer 1

Answer:

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Step-by-step explanation:

let 'a' be any positive integer.

on dividing a by b, we get q and r as quotient and remainder

a=bq+r,

put b=3

a=3q+r. [r=0,1,2]

so, a=3q+0=3q. or. a=3q+1. or a=3q+2

if a=3q then a²=(3q)²=9q²=3(3q²)=3p

if a=3q+1 then a²=(3q+1)²

{use formula (a+b)²=a²+2ab+b²}

=(3q)²+2(3q)(1)+1

=9q+6q+1

=3(3q²+2q)+1

=3p+1

if a=3q+2, then a²=(3q+2²)

=(3q)²+2(3q)(2)+2²

=9q²+12q+4

=9q²+12q+3+1

=3(3q²+4q+1)+1

=3p+1

therefore, square of any positive integer can either be of the form 3p or 3p+1 for some integer 'p'

Answer 2

Answer:

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