Question

There are two consecutive odd integers, whose product is $ 99 $ . Find the integers.

Answer 1

Hint: Odd numbers are those numbers which are not divisible by 2 and when it is divided by $ 2 $ it gives the remainder $ 1 $ always. Example, $ 1,3,5,7,9,11,13,15,17,19,........ $ , here, these numbers follow each other so, it is called the consecutive odd integer and we also have to notice that the ones digit of the odd numbers are among $ 1,3,5,7 $ and $ 9 $ .

Complete step by step solution:
Let the first odd integer is $ 2x - 1 $ and the second odd integer is $ 2x + 1 $ , as we know that it is not divisible by $ 2 $ that’s why we have taken the number of the multiple of $ 2 $ by adding and subtracting 1, as the numbers are consecutive and we have also given that,
First number × Second number $ = 99 $
Now, let us substitute the first number and the second number in the above formed equation. After substituting we get,
 $ \left( {2x - 1} \right)\left( {2x + 1} \right) = 99 $
On further solving, we get,
\[
   \Rightarrow 4{x^2} - 1 = 99 \\
   \Rightarrow 4{x^2} = 99 + 1 \\
   \Rightarrow 4{x^2} = 100 \\
   \Rightarrow {x^2} = \dfrac{{100}}{4} \\
   \Rightarrow {x^2} = 25 \\
   \Rightarrow x = \sqrt {25} \\
   \Rightarrow x = \pm 5 \;
 \]
Now, we will substitute $ + 5 $ , to find the numbers then first number is,
 $
   = 2x - 1 \\
   = 2 \times 5 - 1 \\
   = 10 - 1 \\
   = 9 \;
  $
Second number is,
 $
   = 2x + 1 \\
   = 2 \times 5 + 1 \\
   = 10 + 1 \\
   = 11 \;
  $
And if we substitute $ - 5 $ , to find the numbers then first number is,
 $
   = 2x - 1 \\
   = 2 \times \left( { - 5} \right) - 1 \\
   = - 10 - 1 \\
   = - 11 \;
  $
Second number is,
 $
   = 2x + 1 \\
   = 2 \times \left( { - 5} \right) + 1 \\
   = - 10 + 1 \\
   = - 9 \;
  $
Hence, the consecutive integers are $ 9,11 $ and $ - 9, - 11 $ .
So, the correct answer is “ $ 9,11 $ and $ - 9, - 11 $ ”.

Note: We can solve this problem by just thinking of the numbers, if the question is a multiple choice question then we don’t need to solve it properly, we can just think of the two consecutive odd integers whose product is $ 99 $ . So, by thinking we can get our answer easily that $ 9 $ and $ 11 $ are those odd numbers whose product is $ 99 $ .