Question

Show that every odd integer of the form (4q+1) (4q+3) where qis some integer

Answer 1

Answer:

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