The A.M of two numbers is 17 and their G.M. is 8. Find the numbers.

Explanation

To solve the problem, we need to find two numbers \( A \) and \( B \) given that their Arithmetic Mean (A.M.) is 17 and their Geometric Mean (G.M.) is 8.

### Step-by-step Solution:

1. **Understanding A.M. and G.M.**:
- The Arithmetic Mean (A.M.) of two numbers \( A \) and \( B \) is given by:
\[ \text{A.M.} = \frac{A + B}{2} \]
- The Geometric Mean (G.M.) of two numbers \( A \) and \( B \) is given by:
\[ \text{G.M.} = \sqrt{A \cdot B} \]

2. **Setting up the equations**:
- From the problem, we know:
\[ \frac{A + B}{2} = 17 \]
Multiplying both sides by 2 gives:
\[ A + B = 34 \quad \text{(Equation 1)} \]
- For the G.M., we have:
\[ \sqrt{A \cdot B} = 8 \]
Squaring both sides gives:
\[ A \cdot B = 64 \quad \text{(Equation 2)} \]

3. **Expressing one variable in terms of the other**:
- From Equation 1, we can express \( B \) in terms of \( A \):
\[ B = 34 - A \]

4. **Substituting into the second equation**:
- Substitute \( B \) in Equation 2:
\[ A \cdot (34 - A) = 64 \]
- Expanding this gives:
\[ 34A - A^2 = 64 \]
- Rearranging this leads to:
\[ A^2 - 34A + 64 = 0 \quad \text{(Equation 3)} \]

5. **Solving the quadratic equation**:
- We can solve Equation 3 using the factorization method. We need two numbers that multiply to \( 64 \) and add up to \( 34 \). The numbers are \( 32 \) and \( 2 \):
\[ A^2 - 32A - 2A + 64 = 0 \]
- Factoring gives:
\[ (A - 32)(A - 2) = 0 \]
- Thus, the solutions for \( A \) are:
\[ A = 32 \quad \text{or} \quad A = 2 \]

6. **Finding corresponding values of \( B \)**:
- If \( A = 32 \):
\[ B = 34 - 32 = 2 \]
- If \( A = 2 \):
\[ B = 34 - 2 = 32 \]

7. **Conclusion**:
- The two numbers are \( 32 \) and \( 2 \).

### Final Answer:
The numbers are \( 32 \) and \( 2 \).