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Question

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

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Solution

Concept : 1 Mark
Application : 1 Mark
Proof : 1 Mark

Let us start with taking a, where a is a positive odd integer.

We apply the division algorithm with a and b=4

Since 0r<4 , the possible remainders are 0, 1, 2 and 3.

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

However, since a is odd, a cannot be 4q, 4q+2 (since they are both divisible by 2).

Any odd integer is of the form 4q+1, 4q+3


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