Question

How do you find three consecutive odd integers, whose sum is 57?

Answer 1

Hint: The odd integers are written in the form $2x + 1,\,x \in N$ where N are the natural numbers. Take three integers as the form $x,\,x + 2,x + 4$ add them and make them equal to the given number. Find x and other odd integers.

Complete step by step solution:
The objective of the problem is to find the three consecutive odd integers whose sum is 57.
About Odd numbers: Odd numbers are whole numbers which when divided by 2 gives remainder as one. Odd numbers are written in the form $2x + 1,\,x \in N$ where N is the natural numbers. The odd integers are 1,3,5,7,9,11, and so on. One is the first positive odd integer.
Given that the sum of three consecutive odd integers is 57.
Let the required one integer be x. We need three integers so one integer is $x$ next odd integer can be written as $x + 2$ and the third integer can be written as $x + 4$
It is given that the sum of three consecutive integers is 57.
Now we can write
 $\left( x \right) + \left( {x + 2} \right) + \left( {x + 4} \right) = 57$
Add all the terms that are on left hand side , we get
$\left( {3x + 6} \right) = 57$
Taking 3 as common we get
$3\left( {x + 2} \right) = 57$
Diving with three on both sides we get
$
  \dfrac{{3\left( {x + 2} \right)}}{3} = \frac{{57}}{3} \\
  x + 2 = 19 \\
 $
Subtract two on both sides we get
$
  x + 2 - 2 = 19 - 2 \\
  x = 17 \\
 $
Now substitute the x= 17 in $x + 2$ and $x + 4$
\[17 + 2\,,\,\,17 + 4\]
On solving above we get ,
$19,21$
The consecutive odd integers are $17,19,21$.
Now we can check the values by adding them

Therefore, the required three consecutive odd integers are $17,19,21$.

Note:
The sum of two odd numbers is always even. The product of two odd integers is always odd. The sum of an even number and odd number is even and the sum of two odd numbers is odd.