Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.
Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.
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Sol:-
Let us start with taking a, where a is a positive odd integer. We apply the division algorithm with a and b = 4.
Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.
That is, a can be 4q, or 4q + 1, or 4q + 2, or 4q + 3, where q is the quotient.
However, since a is odd, a cannot be 4q or 4q + 2 (since they are both divisible by 2). Therefore, any odd integer is of the form 4q + 1 or 4q + 3.