91Ó°ÊÓ

Consider the statement, "The product of two consecutive odd integers is $143$".

Let the first odd integer be $n$.

Thus, the next consecutive odd integer is $n+2$.

According to given conditions,

$n(n+2)=143$.

Simplifying the quadratic equation, we get,

$$\begin{gathered} n(n+2)=143 \\ {n}^2+2n=143 \\ {n}^2+2n-143=0 \\ {n}^2+13n-11n-143=0 \\ n(n+13)-11(n+13)=0 \\ (n+13)(n-11)=0 \end{gathered}$$

Use the Zero Product Property to set each factor to $0$, we get,

when $n+13=0$,

$n=-13$.

When $n-11=0$,

$n=11$

If the first odd integer is $n=-13$,

then the next odd integer is,

$n+2=-11$.

If the first odd integer is $n=11$,

then the next odd integer is,

$n=13$.

Thus, we get two sets of odd integers,

role="math" localid="1644727267713" $-13,-11$and$11,13$.

Substitute the first set of values,

$$\begin{gathered} n(n+2)=143 \\ -13(-11)=143 \\ 143=143 \end{gathered}$$

This is true.

Substituting the second set of the integers,

$$\begin{gathered} n(n+2)=143 \\ 11\left(13\right)=143 \\ 143=143 \end{gathered}$$

This is true too.

Hence, the two sets of possible integers are$-13,-11$and$11,13$.