Question

The product of two consecutive odd integers is $255$. Find the integers.

Short answer

The two consecutive odd integers are $15,17and-17,-15$.

Step-by-step solution

Step 1. Given Information

There are two consecutive odd integers whose product is $255.$

We have to find the integers.

Step 2. Finding the integers

Let the first integer be $x$

and next consecutive odd integer be $x+2$

The product of two consecutive odd integers is$\left(x\right)(x+2)=255$

Step 3. Solve the equation

The equation we get is

$$\begin{gathered} \left(x\right)(x+2)=255 \\ {x}^2+2x-255=0 \\ {x}^2+17x-15x-255=0 \\ x(x+17)-15(x+17)=0 \\ (x-15)(x+17)=0 \end{gathered}$$

So,

$$\begin{gathered} x-15=0 \\ x=15 \end{gathered}$$and$$\begin{gathered} x+17=0 \\ x=-17 \end{gathered}$$

Step 4. Put the value of x in assumed integers to find the exact integers

If$x=15$

So, the first integer is$15$

And the next consecutive odd integer

is $15+2=17$

If $x=-17$

So, the first integer is $-17$

And the next consecutive odd integer

is$-17+2=-15$

Step 5. Verify the integers

As the product of two consecutive odd integers is$255.$

Let's verify the integers we get by multiplying them.

So, the two consecutive odd integers are $15,17$and their product is role="math" localid="1644588639588" $15\times 17=255.$

And another two consecutive odd integers are $-17,-15$and their product is role="math" localid="1644588756720" $(-17)\times (-15)=255.$

Thus, both pairs of consecutive odd integers are solutions.