Question

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

Answer 1

Let a be any positive integer

by EDL a = bq +r

0 ≤ r < b

possible remainders are 0, 1, 2 , 3

this shows that a can be in the form of 4q, 4q+1, 4q+2, 4q+3 q is quotient

as a is odd a can't be the form of 4q or 4q+2 as they are even

so a ill be in the form of 4q + 1 or 4q+3

hence proved