If three times the larger of the two numbers is divided by the smaller, we get the quotient 6 and remainder 6. If five times the smaller is divided by the larger, we get quotient 2 and remainder 3. Find the numbers.

Answer

To solve the problem, we need to set up equations based on the information given in the question. Let's denote the larger number as \( x \) and the smaller number as \( y \).

### Step 1: Set up the first equation
According to the first part of the question, when three times the larger number is divided by the smaller number, we get a quotient of 6 and a remainder of 6. This can be expressed mathematically as:

\[ 3x = 6y + 6 \]

### Step 2: Simplify the first equation
We can simplify this equation:

\[ 3x - 6y = 6 \]

Dividing the entire equation by 3 gives us:

\[ x - 2y = 2 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
Now, according to the second part of the question, when five times the smaller number is divided by the larger number, we get a quotient of 2 and a remainder of 3. This can be expressed as:

\[ 5y = 2x + 3 \]

### Step 4: Rearrange the second equation
We can rearrange this equation as follows:

\[ 5y - 2x = 3 \quad \text{(Equation 2)} \]

### Step 5: Solve the equations simultaneously
We now have a system of two equations:

1. \( x - 2y = 2 \)
2. \( 5y - 2x = 3 \)

We can solve these equations using substitution or elimination. Let's use substitution. From Equation 1, we can express \( x \) in terms of \( y \):

\[ x = 2y + 2 \]

### Step 6: Substitute \( x \) in Equation 2
Now, substitute \( x \) in Equation 2:

\[ 5y - 2(2y + 2) = 3 \]

### Step 7: Simplify the equation
Expanding this gives:

\[ 5y - 4y - 4 = 3 \]

Combining like terms results in:

\[ y - 4 = 3 \]

### Step 8: Solve for \( y \)
Adding 4 to both sides gives:

\[ y = 7 \]

### Step 9: Find \( x \)
Now that we have \( y \), we can find \( x \) using the expression we derived earlier:

\[ x = 2(7) + 2 = 14 + 2 = 16 \]

### Final Answer
Thus, the larger number \( x \) is 16 and the smaller number \( y \) is 7.

### Summary of the Solution
- Larger number \( x = 16 \)
- Smaller number \( y = 7 \)