Question

show that the square of odd positive integer is either of the form 6 m + 1 or 6 m + 3 for some integer m​

Answer 1

let a be any odd integer and b = 6

then 0 is smaller than or equal to r and r is smaller than b

therefore r = 0,1,2,3,4,5

a=6q+1 for some integer q

a^2 = (6 q +1)^2

=a^2 = 36q^2 + 12q + 1

= 6q(6q + 2) + 1

= 6m +1 .... m is any integer

a= 6q + 2

a^2= (6q + 2)^2

=36q^2+ 24q + 4

=6q ( 6q + 4) + 3+1

= 6q (6q+ 4 +1) +3

= 6m + 3. .....m is any integer