Question

# The product of two consecutive odd integers is equal to $675$. Find the two integers.

Hint: Consecutive odd integers have a difference of $2$ between them. So, consider the two consecutive odd integers to be ${\text{x}}$ and ${\text{x + 2}}$. Put the product of these two integers equal to $675$ and then simplify the equation to find the value of ${\text{x}}$. Put the value of ${\text{x}}$ in the consecutive numbers you considered and you’ll get the answer.

Given, the product of two consecutive integers is equal to $675$. We have to find two integers. Since there is a difference of $2$ between two consecutive odd integers so we can consider the first integer to be ${\text{x}}$ and the second integer to be ${\text{x + 2}}$. Then according to question,
$\Rightarrow {\text{x}} \times \left( {{\text{x + 2}}} \right) = 675 \\ \Rightarrow {{\text{x}}^2} + 2{\text{x = 675}} \\$
If we add one on both sides we can make a perfect square which will make solving the equation easier. So add $1$ on both the sides-
$\Rightarrow {{\text{x}}^2} + 2{\text{x + 1 = 675 + 1}} \\ \Rightarrow {\left( {{\text{x}} + 1} \right)^2} = 676 \\$
Now remove the square from ${\text{x + 1}}$
$\Rightarrow {\text{x + 1 = }}\sqrt {676} = 26$
Now transfer 1 from left side to the right side of equation and we have,
$\Rightarrow {\text{x = 26 - 1 = 25}}$
Now we have the value of${\text{x}}$, so on putting the value in the considered integers we get-
${\text{x}} = 25 \\ {\text{x}} + 2 = 27 \\$
Here since the product of these two integers is positive so both integers can either be positive or negative as $\left( - \right) \times \left( - \right) = +$ and $\left( + \right) \times \left( + \right) = +$
Hence the two integers are either $25{\text{and}}27$ or$- 25{\text{ and - 27}}$.

Note: Consecutive integers are the integers that are continuous like one after other in a series. Consecutive odd integers are continuous integers which are odd and they have a difference of 2 between them. They are of form $2{\text{n + 1,2n + 3}}$ and so on.