Question

The product of three consecutive odd numbers is $9177$ . What is their sum?

Answer 1

Hint: Here we have been given the product of three consecutive odd numbers and we have to find the sum of those three numbers. Firstly we will let those numbers be such that the difference between consecutive numbers is $2$ . Then we will form an equation by multiplying them and putting them equal to the value given. Finally we will solve the equation and get the three numbers and add them to get the desired answer.

Complete step by step solution:
The product of three consecutive numbers is given as:
$9177$
Let the three numbers be
$x-2,x,x+2$…..$\left( 1 \right)$
So we get the equation as:
$\left( x-2 \right)\times x\times \left( x+2 \right)=9177$
$\Rightarrow x\times \left( x\times x+x\times 2-2\times x-2\times 2 \right)=9177$
On simplifying them further we get,
$\Rightarrow x\times \left( {{x}^{2}}-4 \right)=9177$
$\Rightarrow {{x}^{3}}-4x=9177$….$\left( 2 \right)$
Now as know that $8000<9177<9261$ which is same as ${{20}^{3}}<9177<{{21}^{3}}$
So we can let $x=21$ and check whether it satisfies the equation (2) as follows:
$\Rightarrow {{\left( 21 \right)}^{3}}-4\times 21=9177$
$\Rightarrow 9261-84=9177$
So we get,
$\Rightarrow 9177=9177$
Hence the equation is satisfied which means
$x=21$
Substitute the above value in equation (1) we get,
$21-2,21,21+2$
$\Rightarrow 19,21,23$
So we get the three consecutive odd terms as $19,21,23$
Sum of three consecutive numbers $=19+21+23$
Sum of three consecutive numbers $=63$
Hence sum of three consecutive odd numbers whose product is $9177$ is $63$
So, the correct answer is “63”.

Note: Consecutive terms are those that come one after the other and have a difference of one in this case they were odd consecutive terms and hence the difference among them is two. It was easy to solve when two numbers we let were the same with different middle signs and the cube power was not there to deal with. We can cross check our answer as $19\times 21\times 23=9177$ and which satisfies the given condition. Odd terms are those which are not divisible by $2$ .In this type of question we have to decide where the given product value falls according to the highest power of the unknown variable.