Show that any positive odd integer is of the form 6q + 1, or 6q + 3 or

1.	Show that any positive odd integer is of the form 6q + 1, or 6q + 3 or 6q + 5, where q is some integer.
Answer» Let a is a positive odd integer and apply Euclid’s division algorithm a = 6q + r, Where 0 ≤ r < 6 for 0 ≤ r < 6 probable remainders are 0, 1, 2, 3, 4 and 5. a = 6q + 0 or a = 6q + 1 or a = 6q + 2 or a = 6q + 3 or a = 6q + 4 or a = 6q + 5 may be form Where q is quotient and a = odd integer. This cannot be in the form of 6q, 6q + 2, 6q + 4. [all divides by 2] Hence, any positive odd integer is of the form 6q + 1, or 6q + 3 or (6q + 5)

Show that any positive odd integer is of the form 6q + 1, or 6q + 3 or 6q + 5, where q is some integer.

Answer»

Let a is a positive odd integer and apply Euclid’s division algorithm

a = 6q + r, Where 0 ≤ r < 6

for 0 ≤ r < 6 probable remainders are 0, 1, 2, 3, 4 and 5.

a = 6q + 0

or a = 6q + 1

or a = 6q + 2

or a = 6q + 3

or a = 6q + 4

or a = 6q + 5 may be form

Where q is quotient and a = odd integer.

This cannot be in the form of 6q, 6q + 2, 6q + 4.

[all divides by 2]

Hence, any positive odd integer is of the form 6q + 1, or 6q + 3 or (6q + 5)