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 be any positive odd integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r, for some integer q ≥ 0 and 0 ≤ r < 6. i.e., the possible remainders are 0, 1, 2, 3, 4, 5. Thus, a can be of the form 6q, or 6q + 1, or 6q + 2, or 6q + 3, or 6q + 4, or 6q + 5, where q is some quotient. Since a is an odd integer, so a cannot be of the form 6q, or 6q + 2, or 6q +4 (since they are even). Thus, a is of the form 6q + 1, 6q + 3, or 6q + 5, where q is some integer. Hence, any odd positive integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where q is some integer.

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 be any positive odd integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r, for some integer q ≥ 0 and 0 ≤ r < 6.
i.e., the possible remainders are 0, 1, 2, 3, 4, 5.
Thus, a can be of the form 6q, or 6q + 1, or 6q + 2, or 6q + 3, or 6q + 4, or 6q + 5, where q is some quotient.
Since a is an odd integer, so a cannot be of the form 6q, or 6q + 2, or 6q +4 (since they are even).
Thus, a is of the form 6q + 1, 6q + 3, or 6q + 5, where q is some integer.

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

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