Find the greatest number that can divide 76 and 60 leaving remainders of 4 and 6, respectively.

Answer

To find the greatest number that can divide 76 and 60 leaving remainders of 4 and 6 respectively, we can follow these steps:

### Step 1: Adjust the numbers based on the remainders
We need to adjust the numbers 76 and 60 by subtracting their respective remainders.

- For 76, subtract the remainder 4:
\[ 76 - 4 = 72 \]

- For 60, subtract the remainder 6:
\[ 60 - 6 = 54 \]

### Step 2: Find the Highest Common Factor (HCF)
Now, we need to find the HCF of the adjusted numbers 72 and 54.

**Factors of 72:**
- 72 can be factored as:
\[ 72 = 2^3 \times 3^2 \]

**Factors of 54:**
- 54 can be factored as:
\[ 54 = 2^1 \times 3^3 \]

### Step 3: Identify the common factors
To find the HCF, we take the lowest power of each common prime factor.

- For the prime factor 2:
- In 72, the power is 3.
- In 54, the power is 1.
- The minimum is \(2^1\).

- For the prime factor 3:
- In 72, the power is 2.
- In 54, the power is 3.
- The minimum is \(3^2\).

### Step 4: Calculate the HCF
Now, we multiply the common factors together:
\[ HCF = 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Step 5: Verify the result
To ensure that 18 is the correct answer, we can check the division of the original numbers:
- When dividing 76 by 18:
\[ 76 \div 18 = 4 \quad \text{(remainder 4)} \]
- When dividing 60 by 18:
\[ 60 \div 18 = 3 \quad \text{(remainder 6)} \]

Both conditions are satisfied.

### Final Answer
The greatest number that can divide 76 and 60 leaving remainders of 4 and 6 respectively is:
\[ \boxed{18} \]