A number consists of two digits. The product of these digits is 14. If 45 is subtracated from the number, the digits interchange their places. Find the number.

Answer

To solve the problem step by step, we will define the digits of the two-digit number and set up equations based on the information given.

### Step 1: Define the digits
Let the two-digit number be represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

### Step 2: Set up the equations
According to the problem:
1. The product of the digits is 14:
\[ xy = 14 \]
2. If 45 is subtracted from the number, the digits interchange their places:
\[ 10x + y - 45 = 10y + x \]

### Step 3: Simplify the second equation
Rearranging the second equation gives us:
\[ 10x + y - 45 = 10y + x \]
\[ 10x - x + y - 10y = 45 \]
\[ 9x - 9y = 45 \]
Dividing the entire equation by 9:
\[ x - y = 5 \quad \text{(Equation 1)} \]

### Step 4: Express \(y\) in terms of \(x\)
From the product equation \(xy = 14\), we can express \(y\) in terms of \(x\):
\[ y = \frac{14}{x} \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 2 into Equation 1
Substituting \(y\) from Equation 2 into Equation 1:
\[ x - \frac{14}{x} = 5 \]
Multiplying through by \(x\) to eliminate the fraction:
\[ x^2 - 14 = 5x \]
Rearranging gives us a standard form quadratic equation:
\[ x^2 - 5x - 14 = 0 \]

### Step 6: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -5\), and \(c = -14\):
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 + 56}}{2} \]
\[ x = \frac{5 \pm \sqrt{81}}{2} \]
\[ x = \frac{5 \pm 9}{2} \]

Calculating the two possible values for \(x\):
1. \(x = \frac{14}{2} = 7\)
2. \(x = \frac{-4}{2} = -2\) (not valid since \(x\) must be a digit)

Thus, \(x = 7\).

### Step 7: Find \(y\)
Substituting \(x = 7\) back into Equation 2 to find \(y\):
\[ y = \frac{14}{7} = 2 \]

### Step 8: Form the two-digit number
The two-digit number is:
\[ 10x + y = 10(7) + 2 = 70 + 2 = 72 \]

### Final Answer
The number is **72**.