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**Hint:**Start by considering the two numbers as some variables, use the statements given in the question to form the equations. Represent one variable in the form of another variable and substitute it in the other equation and find out the value of one of the variables, substitute this value in any equation to find the other variable.

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

Firstly, let us assume the two numbers be x and y

Then according to the statement given in the question, sum of two number is 27, we will be having equation as

$ x + y = {\text{2}}7 \to eqn.1 $

Similarly, according to the second statement given in the question, we will be having the equation as

$ xy = 182 \to eqn.2 $

Now, the equation 1 can be written as

$ x = {\text{2}}7 - y \to eqn.3 $

Now, we will substitute this value of x in equation 2, we will get

$ \left( {27 - y} \right)\left( y \right) = 182 $

So, the above equation can also be written as

$ {y^2} - 27y + 182 = 0 $

On splitting the middle term ,we will get

$ \left( {y - 13} \right)\left( {y - 14} \right) = 0 $

From the equation above either

$ y - 13 = 0 $ or $ y - 14 = 0 $

which gives us values of y as 13 and 14.

Now, Putting these values of y in equation(1) we will get $ x = 13 $ when $ y = 13 $ and $ x = 14 $ when $ y = 14 $

Hence, the two numbers are 13 and 14, as the first pair of solutions i.e. (13,13) would not give us the sum of 27.

Therefore, 13 and 14 are the two numbers.

**Note:**Similar questions can be solved by using the same procedure as above. Attention must be given while substituting the values as any wrong substitution might lead to a wrong answer. Always check the answer by back substitution in the equations formed by the conditions provided.

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