Question

Use a direct proof to show that the product of two odd numbers is odd.

Short answer

When two odd integers are multiplied together, they produce an odd number.

Step-by-step solution

Introduction

Assume m and n be two odd integers.

The purpose is to show that when two odd integers are multiplied together, they produce an odd number.

It is enough to show that \(m.n\)is odd.

Taking two odd integers and solving for the required result

Since, m and n be two odd integers. Then, \(m = 2k + 1\)and\(n = 2l + 1\), for any integers \(k\) and \(l\).

Now,

\(m.n = \left( {2k + 1} \right)\left( {2l + 1} \right)\)\(\begin{aligned}{*{20}{l}}{}\\{ = 4kl + 2k + 2l + 1}\\{ = 2\left( {2kl + k + l} \right) + 1}\end{aligned}\)

\( \Rightarrow m.n = 2p + 1\), where taking \(p\)as \(2kl + k + l = integer\)

Thus,\(m.n\)is odd.

Therefore, when two odd integers are multiplied together, they produce an odd number.