Q3 of 104 Page 2

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

Let a be any 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.


i.e. a can be 4q, or 4q + 1, or 4q + 2, or 4q + 3, where q is the quotient.


As we know a is odd, a can’t be 4q or 4q + 2 because they both are divisible by 2.


Therefore, any positive odd integer is of the form 4q + 1 or 4q + 3.


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