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# Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3, and 4 respectively, they add up to 74. Find these numbers.

Last updated date: 20th Jun 2024
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Hint: Here, we have to find the three consecutive integers. We will assume the smallest of the three consecutive integers to be $x$. Using the given information, we will form a linear equation in terms of $x$. We will solve the obtained equation to find the value of $x$, and hence, the three consecutive integers.

Let the smallest integer of the three consecutive integers be $x$.
Therefore, the next two consecutive integers will be $x + 1$ and $x + 2$.
First, we will arrange these in increasing order.
Therefore, we get $x$, $x + 1$, and $x + 2$.
Now, we will use the given information to form a linear equation in terms of $x$.
The three consecutive integers are multiplied by 2, 3, and 4 respectively.
Multiplying $x$ by 2, we get $2x$.
Multiplying $x + 1$ by 3, we get $3\left( {x + 1} \right)$.
Multiplying $x + 2$ by 4, we get $4\left( {x + 2} \right)$.
It is given that the three consecutive integers multiplied by 2, 3, and 4 respectively, add up to 74.
Therefore, we can form the equation
$2x + 3\left( {x + 1} \right) + 4\left( {x + 2} \right) = 74$
We will solve this equation to get the value of $x$.
Multiplying the terms of the expression, we get
$\Rightarrow 2x + 3x + 3 + 4x + 8 = 74$
Adding the like terms of the expression, we get
$\Rightarrow 9x + 11 = 74$
Subtracting 11 from both sides of the equation, we get
$\begin{array}{l} \Rightarrow 9x + 11 - 11 = 74 - 11\\ \Rightarrow 9x = 63\end{array}$
Dividing both sides by 9, we get
$\begin{array}{l} \Rightarrow \dfrac{{9x}}{9} = \dfrac{{63}}{9}\\ \Rightarrow x = 7\end{array}$
Therefore, the smallest consecutive integer out of the three numbers is 7.
Substituting $x = 7$ in $x + 1$ and $x + 2$, we get the other two integers as
$x + 1 = 7 + 1 = 8$
$x + 2 = 7 + 2 = 9$
Therefore, the three consecutive integers are 7, 8, and 9.

Note: We have used the distributive property of multiplication to find the products $3\left( {x + 1} \right)$ and $4\left( {x + 2} \right)$. The distributive property of multiplication states that $a\left( {b + c} \right) = a \cdot b + a \cdot c$.
We can verify our answer by multiplying 7, 8, 9 by 2, 3, 4 respectively and checking the sum.
Multiplying 7 by 2, we get 14.
Multiplying 8 by 3, we get 24.
Multiplying 9 by 4, we get 36.
The sum of 14, 24, and 36 is 74.
Hence, we have verified the answer.