Question

Find two consecutive odd numbers whose product is$323$.

Short answer

The pair of consecutive numbers are$-17,-19$and$17,19$.

Step-by-step solution

Step 1. Given information.

Two consecutive odd numbers whose product is$23$.

Step 2. Find two consecutive odd numbers.

Let the first odd number is $x$.

Then the second odd number is $x+2$.

Now, according to the question:

$\begin{array}{rcl}x\left(x+2\right)& =& 323\\ x^2+2x& =& 323\\ x^2+2x-323& =& 0\end{array}$

Step 3. Solve the equation.

$\begin{array}{rcl}{x}^2+2x-323& =& 0\\ x^2+19x-17x-323& =& 0\\ x\left(x+19\right)-17\left(x+19\right)& =& 0\\ \left(x+19\right)\left(x-17\right)& =& 0\end{array}$

Step 4. Apply zero product rule.

$\begin{array}{rcl}x+19& =& 0\\ x& =& -19\end{array}$and role="math" localid="1645768635050" $\begin{array}{rcl}x-17& =& 0\\ x& =& 17\end{array}$

Hence, the first pair of consecutive numbers are $-19$and $-19+2=-17$.

The second pair of consecutive numbers are$17$and$17+2=19$.