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 step by step, we will denote the larger number as \( x \) and the smaller number as \( y \).

### Step 1: Set up the equations based on the problem statement.

From the first condition:
- If three times the larger number is divided by the smaller, we get a quotient of 6 and a remainder of 6.
- This can be expressed as:
\[ 3x = 6y + 6 \]
Rearranging gives us:
\[ 3x - 6y = 6 \quad \text{(Equation 1)} \]

From the second condition:
- If five times the smaller number is divided by the larger, we get a quotient of 2 and a remainder of 3.
- This can be expressed as:
\[ 5y = 2x + 3 \]
Rearranging gives us:
\[ -2x + 5y = 3 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

We will solve these two equations simultaneously.

From Equation 1:
\[ 3x - 6y = 6 \]
We can express \( x \) in terms of \( y \):
\[ 3x = 6y + 6 \implies x = 2y + 2 \quad \text{(Equation 3)} \]

Now, substitute Equation 3 into Equation 2:
\[ -2(2y + 2) + 5y = 3 \]
Expanding this gives:
\[ -4y - 4 + 5y = 3 \]
Combining like terms results in:
\[ y - 4 = 3 \]
Adding 4 to both sides:
\[ y = 7 \]

### Step 3: Find the value of \( x \).

Now that we have \( y \), we can substitute it back into Equation 3 to find \( x \):
\[ x = 2(7) + 2 = 14 + 2 = 16 \]

### Conclusion:

The larger number \( x \) is 16 and the smaller number \( y \) is 7.

Thus, the two numbers are:
- Larger number: \( 16 \)
- Smaller number: \( 7 \)

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