A number consists of two digits. When it id divided by the sum of its digits, the quotient is 6 with no remainder. When the number is diminished by 9, the digits are reversed. Find the number.

Explanation

To solve the problem step by step, we will define the two-digit number and set up equations based on the conditions given in the question.

### Step 1: Define the two-digit number
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 first equation
According to the problem, when the number is divided by the sum of its digits, the quotient is 6 with no remainder. This can be expressed as:
\[ \frac{10x + y}{x + y} = 6 \]
Multiplying both sides by \(x + y\) gives:
\[ 10x + y = 6(x + y) \]
Expanding the right side:
\[ 10x + y = 6x + 6y \]
Rearranging the equation results in:
\[ 10x - 6x = 6y - y \]
This simplifies to:
\[ 4x = 5y \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
The problem states that when the number is diminished by 9, the digits are reversed. This can be expressed as:
\[ 10x + y - 9 = 10y + x \]
Rearranging this gives:
\[ 10x + y - x - 10y = 9 \]
Simplifying leads to:
\[ 9x - 9y = 9 \]
Dividing the entire equation by 9 gives:
\[ x - y = 1 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \(4x = 5y\)
2. \(x - y = 1\)

From Equation 2, we can express \(x\) in terms of \(y\):
\[ x = y + 1 \]

### Step 5: Substitute into Equation 1
Substituting \(x = y + 1\) into Equation 1:
\[ 4(y + 1) = 5y \]
Expanding this gives:
\[ 4y + 4 = 5y \]
Rearranging leads to:
\[ 4 = 5y - 4y \]
Thus:
\[ y = 4 \]

### Step 6: Find \(x\)
Now substituting \(y = 4\) back into Equation 2:
\[ x - 4 = 1 \]
So:
\[ x = 5 \]

### Step 7: Find the two-digit number
Now we have \(x = 5\) and \(y = 4\). The two-digit number is:
\[ 10x + y = 10(5) + 4 = 54 \]

### Conclusion
The two-digit number is **54**.