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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.

**Complete step-by-step answer:**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.

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