Question

How do you find three consecutive odd integers whose sum is $117$?

Answer 1

Hint: Here we must know what consecutive means. This means one after the other number. So we can let the first odd integer as a variable and then we can write the next two odd integers in terms of that variable and accordingly apply the given condition to solve the problem given.

Complete step-by-step answer:
Here we are given to find the three consecutive odd integers whose sum is $117$ and therefore we must know what consecutive integers mean. If we are given three consecutive numbers, this means that these three numbers are continuous or one after the other. For example: $5,6,7$ are three consecutive integers. Now here we need to find the consecutive odd integers. We know that odd integer is two more than the previous odd integer.
Hence we can let the first odd integer as a variable.
Let it be $x$
Next consecutive odd integer will be two more than this variable.
So second consecutive integer $ = (x + 2)$
Next consecutive odd integer will be two more than this second integer obtained.
So third consecutive integer $ = (x + 2) + 2 = x + 4$
Now according to the condition we are given that their sum is $117$
Hence we can write this in the equation form as:
$x + x + 2 + x + 4 = 117$
Now solving this we will get:
$
  3x + 6 = 117 \\
\Rightarrow 3x = 117 - 6 \\
\Rightarrow 3x = 111 \\
\Rightarrow x = \dfrac{{111}}{3} \\
\Rightarrow x = 37 \\
 $
Hence we get the first odd integer as $37$
Second will be$ = x + 2 = 37 + 2 = 39$
Third will be$ = x + 4 = 37 + 4 = 41$
Hence we get the three consecutive odd integers as $37,39,41$

Note: Here we must know that consecutive odd and even integers have the general form $x,x + 2,x + 4,x + 6.........{\text{and so on}}$ where $x$ can be any odd or even integer.
If we are given consecutive integers only, so then we can take the variables to be $x,x + 1,x + 2,........{\text{ and so on}}$