Question

By Euclid's division lemma show that the square of any positive integer is in the form of 3n or 3n+1

Answer 1

Let a be any positive integer
a=1,2,3,4......
b=3
Euclid division lemma:-
a=bn+r
a=3n+r where n≤0 and 0≤r<3
The possible values of r are 0,1 and 2
Take square of any positive integer.
(1)²=1=3(0)+1=3n+1 (r=1)
(2)²=4=3(1)+1=3n+1 (r=1)
(3)²=9=3(3)+0=3n (r=0)
(4)²=16=3(5)+1=3n+1 (r=1)
(5)²=25=3(8)+1=3n+1 (r=1)

So,the square of any positive integer is of the form 3n or 3n+1

Hope it helps