Question

2. Use Euclid's Lemma show that square of any
positive integer is either of the form 3m or 3 + 1
for some integer m.​

Answer 1

Step-by-step explanation:

let x can be any positive integer and y=3 .

by Euclid's lemma then x=3q+r for some integer q>0 and r= 0,1,2 as r>_0 and r<3

therefore x= 3q, 3q+1, 3q+2

Now as per the question given by squaring both the sides , we get

x²= (3q)² =9q²= 3×3q²

let 3q²=m

therefore, x²= 3m.........(1)

x²= (3q+1)²= (3q)²+(1)²+(2)(3q)(1)

9q²+1+6q

3(3q²+2q) +1

substitute, 3q²+2q= m to get

x²= 3m+1............(2)

x²=(3q+2)²= (3q)²+(2)²+(2)(3q)(2)

= 9q²+4+12q= 3(3q²+4q+1) +1

Again substitute, 3q²+4q+1= m , to get

x²= 3m +1 .............(3)

Hence from equation 1,2, and3 , we can say that the square of any positive integer is either of the form 3m or 3m+1 for some integer m.

Answer 2

Answer:

It is the correct answer.

Step-by-step explanation:

Hope this attachment helps you.