Question

4
Use Euclid? diruson limma te showe
that the square of any positive
integer is either of the beard
3mar 3 m+ 1 far same integer m.
​

Answer 1

Let 'a' be any positive integer

On dividing it by 3, let 'q' be the quotient and 'r' be the remainder.

Such that,

a = 3q+r , where r = 0,1,2

when r = 0

a = 3q

when r = 1

a = 3q + 1

when r = 2

a = 3q + 2

when,a = 3q

squaring both sides

a^2 = 9q^2

a^2 = 3 (3q^2)

a^2 = 3m

where m = 3q^2

When a = 3q +1

squaring both sides

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

a^2 = 9q^2 + 6q + 1

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

a^2 = 3m+1

where m = 3q^2 + 2q

when a = 3q + 2

squaring both sides

a^2 = (3q+2)^2

a^2 = 9q^2 +12q + 4

a^2 = 9q^2 + 12q + 3 + 1

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

a^2 = 3m+1

where m = 3q^2+ 4q+1

Therefore, the sum of any positive integer is either in the form of 3m or 3m+1

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