Question

Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is an integer.

Answer 1

As per Euclid’* division lenima

If a and b are two positive integers, then a = bq + r where 0 ≤ r < b

Let positive integer be a and b = 4.

Hence a = 4q + r

Where (0 ≤ r < 4)

r is an integer greater than or equal to 0 and less than 4.

Hence r can be either 0, 1, 2, or 3.

If r = 1

a = 4q + 1

This will always be an odd integer.

If r = 3

a = 4q + 3

This will always be an odd integer. Therefore any odd integer is of the form 4q + 1 or 4q + 3 Hence proved.