Find the greartest number that divides 699 and 572 leaving remainders 6 and5 respectively .

Answer

To find the greatest number that divides 699 and 572 leaving remainders of 6 and 5 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
We need to subtract the remainders from the given numbers to find the numbers that will be completely divisible by the greatest number we are looking for.

- For 699, subtract 6:
\[ 699 - 6 = 693 \]

- For 572, subtract 5:
\[ 572 - 5 = 567 \]

### Step 2: Find the HCF (Highest Common Factor) of the adjusted numbers
Now we need to find the HCF of 693 and 567.

#### Factorization of 693:
1. Divide by 3:
\[ 693 \div 3 = 231 \]
2. Divide 231 by 3:
\[ 231 \div 3 = 77 \]
3. Divide 77 by 7:
\[ 77 \div 7 = 11 \]
4. 11 is a prime number.

So, the prime factorization of 693 is:
\[ 693 = 3^2 \times 7 \times 11 \]

#### Factorization of 567:
1. Divide by 3:
\[ 567 \div 3 = 189 \]
2. Divide 189 by 3:
\[ 189 \div 3 = 63 \]
3. Divide 63 by 3:
\[ 63 \div 3 = 21 \]
4. Divide 21 by 3:
\[ 21 \div 3 = 7 \]
5. 7 is a prime number.

So, the prime factorization of 567 is:
\[ 567 = 3^4 \times 7 \]

### Step 3: Identify the common factors
Now we identify the common factors from the factorizations:
- From 693: \(3^2\) and \(7\)
- From 567: \(3^4\) and \(7\)

The common factors are:
- \(3^2\) (the minimum power of 3)
- \(7^1\)

### Step 4: Calculate the HCF
Now we multiply the common factors to find the HCF:
\[ HCF = 3^2 \times 7 = 9 \times 7 = 63 \]

### Conclusion
The greatest number that divides 699 and 572 leaving remainders of 6 and 5 respectively is:
\[ \boxed{63} \]