Question

Use Euclid's division lemma to show that the squar of any postive integer is of the form 3p, 3p+1

Answer 1

HEYA MATE HERE U GO.

YOUR REQUIRED ANSWER:)

->square of any positive integer = 3p or 3p + 1

let a be any positive integer and b=3

0<=r<b

0<=r<3

r=0,1,2

.....(I) a=bq+r

a=3q+r

if r=3q+0 in equation... (I)

a=3q

squaring both side.

a² = 9q²

a² = 3 x (3q²)

a² = 3p [where p= 3q²]

if r= 3q+1 in equation.. (I)

a= 3q+1

squaring both side

a²=(3q+1)²

a²=(3q) ²+(1)²+2(3q)(1)

a²= 9q² + 1+ 6q

a²= 9q² + 6q+1

|_________|

m

a²= 3(3q²+2q) +1

a²= 3m +1 [where m= (3q²+2q²).

hence, proved.

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