The greatest number that leaves 6 on division on 5193 or 5856 is.

Options

A. 21
B. 29
C. 39
D. 37

Answer

To solve the problem of finding the greatest number that leaves a remainder of 6 when dividing 5193 and 5856, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers and Remainder**:
- The numbers given are 5193 and 5856.
- The remainder we want when dividing these numbers is 6.

2. **Adjust the Numbers**:
- To find the numbers that are fully divisible by our desired greatest number, we need to subtract the remainder from both numbers.
- For 5193:
\[ 5193 - 6 = 5187 \]
- For 5856:
\[ 5856 - 6 = 5850 \]

3. **Find the HCF (Highest Common Factor)**:
- Now, we need to find the HCF of the adjusted numbers 5187 and 5850.
- We can use the factorization method to find the HCF.

4. **Factorization of 5187**:
- To factor 5187, we can divide it by prime numbers:
\[ 5187 = 39 \times 133 \]
- Further factor 133:
\[ 133 = 7 \times 19 \]
- Therefore, the complete factorization of 5187 is:
\[ 5187 = 3 \times 13 \times 7 \times 19 \]

5. **Factorization of 5850**:
- Now, we factor 5850:
\[ 5850 = 39 \times 150 \]
- Further factor 150:
\[ 150 = 3 \times 5 \times 10 = 3 \times 5 \times 2 \times 5 \]
- Therefore, the complete factorization of 5850 is:
\[ 5850 = 2 \times 3 \times 5^2 \times 39 \]

6. **Identify Common Factors**:
- The common factors of 5187 and 5850 from their factorizations are:
- Both have 39 as a common factor.

7. **Conclusion**:
- Thus, the greatest number that leaves a remainder of 6 when dividing both 5193 and 5856 is:
\[ \text{HCF} = 39 \]

### Final Answer:
The greatest number that leaves a remainder of 6 when dividing 5193 and 5856 is **39**.

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