Question

use uceld,s division lema to show that any square pistive integer is of the form 3p,3p+1

Answer 1

let 'a'be any positive integer and b =3
Then by applying euclid' s division lemma we get,
a=3q+r, where 0 less than or equal to r and r less than b.
Then by above we get,
a=3q or a=3q+1 or a=3q+2
now,
a=3q
a^2= (3q)^2
a^2=9q^2
a^2=3p, where p=3q^2.....................1
now,
a=3q+1
a^2= (3q+1)^2
a^2=9q^2+1+6q
a^2=3q(3q+2)+1
a^2=3p+1, where p=q(3q+2)..................2
now,
a=3q+2
a^2=(3q+2)^2
a^2=9q^2+4+12q
a^2=9q^2+3+1+12q
a^2=3(3q^2+4q+1)+1
a^2=3p+1, where p=(3q^2+4q+1).............. 3
By equation 1,2,3,
we can say that any positive integer is of the form 3p, 3p+1

Answer 2

I hope this answer is helpful 2 u
from above we can conclude that 3p ,3p+1 is any positive integer