Question

Show that the square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer q

Answer 1

Let take a as any positive integer and b = 6.�

Then using Euclid�s algorithm we get�a = 6q + r�here r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2, 3, 4, 5 because 0 <= r < b�and the value of b is 6�

So total possible forms�will�6q+0 , 6q+1 , 6q+2,6q+3,6q+4,6q+5

6q+0�6 is divisible by 2 so it is a even number�

6q+1�6 is divisible by 2 but 1 is not divisible by 2 so it is a odd number�

6q+2�6 is divisible by 2 and 2 is also divisible by 2 so it is a even number�

6q+3�6 is divisible by 2 but 3 is not divisible by 2 so it is a odd number�

6q+4�6 is divisible by 2 and 4 is also divisible by 2 it is a even number

6q+5�6 is divisible by 2 but 5 is not divisible by 2 so it is a odd number

So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5