Find the greatest number which on dividing 391 and 318 leaves remainder 7 and 6 respectively.
A . 20
B. 23
C. 24
D. 32

Options

A. C
B. B
C. A
D. D

Answer

To find the greatest number which, when dividing 391 and 318, leaves remainders of 7 and 6 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
- For 391, subtract the remainder 7:
\( 391 - 7 = 384 \)
- For 318, subtract the remainder 6:
\( 318 - 6 = 312 \)

### Step 2: Find the greatest common divisor (GCD) of the adjusted numbers
We need to find the GCD of 384 and 312. We can do this using the prime factorization method.

### Step 3: Prime factorization of 384
- Divide by 2:
\( 384 \div 2 = 192 \)
- Divide by 2:
\( 192 \div 2 = 96 \)
- Divide by 2:
\( 96 \div 2 = 48 \)
- Divide by 2:
\( 48 \div 2 = 24 \)
- Divide by 2:
\( 24 \div 2 = 12 \)
- Divide by 2:
\( 12 \div 2 = 6 \)
- Divide by 2:
\( 6 \div 2 = 3 \)
- Finally, divide by 3:
\( 3 \div 3 = 1 \)

So, the prime factorization of 384 is:
\( 384 = 2^7 \times 3^1 \)

### Step 4: Prime factorization of 312
- Divide by 2:
\( 312 \div 2 = 156 \)
- Divide by 2:
\( 156 \div 2 = 78 \)
- Divide by 2:
\( 78 \div 2 = 39 \)
- Divide by 3:
\( 39 \div 3 = 13 \)
- Finally, divide by 13:
\( 13 \div 13 = 1 \)

So, the prime factorization of 312 is:
\( 312 = 2^3 \times 3^1 \times 13^1 \)

### Step 5: Find the GCD using the lowest powers of common prime factors
The common prime factors are:
- For 2: the minimum power is \( 2^3 \)
- For 3: the minimum power is \( 3^1 \)

Thus, the GCD is:
\( GCD = 2^3 \times 3^1 = 8 \times 3 = 24 \)

### Step 6: Conclusion
The greatest number which divides both 391 and 318 leaving the specified remainders is \( 24 \).

### Final Answer
The answer is \( 24 \) (Option C).

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