Question

The sum of three consecutive integers is \[75\]. What are the integers?

Answer 1

Hint: Consecutive integers have a difference of one. We will consider one of the integers as \[x\], then the next two integers can be found out by adding one to \[x\] and adding two to \[x\]. Then we will equate the sum of those three integers to the given sum that is \[75\], and find the value of \[x\]. From the value \[x\], we will find the value of the other two integers.

Complete step-by-step answer:
Let, one of the integers is \[ = x\]
Then the next integer is \[ = x + 1\]
Now we can find the third integer by adding one to \[x + 1\], or by directly adding two to \[x\].
\[\therefore \] the third integer \[ = x + 2\]
Now, the sum of the three integers is \[ = x + \left( {x + 1} \right) + \left( {x + 2} \right)\]
But, in the question it is given that the sum of the three integers is equal to \[75\].
So, we will equate both of them as they are equal. So, we get;
\[ \Rightarrow x + \left( {x + 1} \right) + \left( {x + 2} \right) = 75\]
On further simplification we get;
\[ \Rightarrow 3x + 3 = 75\]
Now we will shift \[3\] to RHS. So, we get;
\[ \Rightarrow 3x = 72\]
\[ \Rightarrow x = 24\]
So, one of the numbers is \[24\].
\[\therefore \] the second integer is \[ = 24 + 1 = 25\]
\[\therefore \] the third integer is \[ = 24 + 2 = 26\]
So, the three consecutive integers are \[24,25,26\].
So, the correct answer is “\[24,25,26\]”.

Note: If in the question sum of the three consecutives even numbers is given then if we consider one of the numbers as \[x\], then the other two numbers will be \[x + 2,x + 4\] respectively. Same pattern will be followed for odd integers. One thing to note here is that, we cannot predict any such pattern for prime numbers.