Chapter 1.4, Problem 15GP

To determine

Whether the given conjecture is true. If not, then give a counterexample.

Answer to Problem 15GP

  $\text{Not true}$

Explanation of Solution

Given information:

The product of odd numbers is always even.

Concept Used:

Commutative property:

  $a+b=b+a$

  $a\cdot b=b\cdot a$

Associative property:

  $\left(a+b\right)+c=a+\left(b+c\right)$

  $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

Distributive property:

  $a\left(b+c\right)=ab+ac$

  $\left(b+c\right)a=ba+ca$

Additive identity:

  $a+0=0+a=a$

Multiplicative identity:

  $a\cdot 1=1\cdot a=a$

Zero property of Multiplication:

  $a\cdot 0=0\cdot a=0$

Calculation:

The product of odd numbers is always odd. For example,

  $\begin{array}{c}1\times 3=3,\\ 5\times 9=45,\\ 3\times 5\times 9=135,\\ 11\times 7=77,\\ 7\times 13=91\text{etc}\mathrm{..}\end{array}$

While multiplying two or more odd numbers, the product will always be an odd number as well.

Thus, the conjecture that the product of odd numbers is always evenis not true.