Question

show that the square of any positive intigers is of the form 3m or 3m + 1

Answer 1

Let a and b be any two +ve integers where b=3 ...
so, according to Euclid's Division Lemma:
a=bq+r(where 0 less than or equals to r<b..
So, a=3q+r
Possible values of r=0,1,2..
possible values of a=3q, 3q+1,3q+2...
so case 1:
a=3q
Squaring both sides:
a²=(3q) ²
a²=9q²
a²=9m(where q²=m)
Case -2:
a=(3q+1)
S. B. S. :
a²=(3q+1)²
a²=(3q) ²+(1)²+2(3q)(1)
a²=9q²+1+6q
a²=3(3q²+2q) +1
a²=3m+1 (where 3q²+2q=m)
Case:3:::
a²=(3q+2)²
a²=(3q) ²+(2)²+2(3q)(2)
a²=9q²+4+12q..
a²=9q²+12q+3+1
a²=3(3q²+4q+1)+1
a²=3m+1(where 3q²+4q+1=m)
So, the square of any integer is of the form 3m or 3m+1 for some integer m...