Find the largest number which can divided x,y,z exactly

Answer

To find the largest number that can divide the numbers \( x \), \( y \), and \( z \) exactly, we need to determine the highest common factor (HCF) of these three numbers. Here’s how to do it step by step:

### Step-by-Step Solution

1. **Identify the Numbers**:
- Let’s denote the three numbers as \( x \), \( y \), and \( z \).

2. **Find the Prime Factorization**:
- Factor each of the numbers into their prime factors.
- For example, if \( x = 12 \), \( y = 18 \), and \( z = 24 \):
- \( 12 = 2^2 \times 3^1 \)
- \( 18 = 2^1 \times 3^2 \)
- \( 24 = 2^3 \times 3^1 \)

3. **Identify Common Prime Factors**:
- Look for the prime factors that are common to all three numbers.
- In our example, the common prime factors are \( 2 \) and \( 3 \).

4. **Determine the Lowest Power of Common Factors**:
- For each common prime factor, take the lowest power that appears in the factorizations.
- For \( 2 \): The powers are \( 2^2 \) (from 12), \( 2^1 \) (from 18), and \( 2^3 \) (from 24). The lowest power is \( 2^1 \).
- For \( 3 \): The powers are \( 3^1 \) (from 12), \( 3^2 \) (from 18), and \( 3^1 \) (from 24). The lowest power is \( 3^1 \).

5. **Multiply the Lowest Powers Together**:
- Multiply the lowest powers of all common prime factors to find the HCF.
- In our example:
\[ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

6. **Conclusion**:
- The largest number that can divide \( x \), \( y \), and \( z \) exactly is the HCF, which in this case is \( 6 \).

### Final Answer:
The largest number that can divide \( x \), \( y \), and \( z \) exactly is the HCF, which is \( 6 \).