The greatest number that will divide 729 and 901 leaving remainders 9 and 5 respectively is:

Options

A. 15
B. 16
C. 19
D. 20

Answer

To find the greatest number that will divide 729 and 901 leaving remainders of 9 and 5 respectively, we can follow these steps:

### Step 1: Set up the equations
Let the greatest number be \( n \). According to the problem, when 729 is divided by \( n \), the remainder is 9. This can be expressed as:
\[ 729 \equiv 9 \mod n \]
This implies:
\[ 729 - 9 = 720 \text{ is divisible by } n \]

Similarly, when 901 is divided by \( n \), the remainder is 5:
\[ 901 \equiv 5 \mod n \]
This implies:
\[ 901 - 5 = 896 \text{ is divisible by } n \]

### Step 2: Find the numbers to consider
From the above, we need to find the greatest number \( n \) that divides both 720 and 896.

### Step 3: Factorize the numbers
Now we will factorize both 720 and 896.

**Factorization of 720:**
1. Divide by 2: \( 720 \div 2 = 360 \)
2. Divide by 2: \( 360 \div 2 = 180 \)
3. Divide by 2: \( 180 \div 2 = 90 \)
4. Divide by 2: \( 90 \div 2 = 45 \)
5. Divide by 3: \( 45 \div 3 = 15 \)
6. Divide by 3: \( 15 \div 3 = 5 \)
7. Divide by 5: \( 5 \div 5 = 1 \)

So, the prime factorization of 720 is:
\[ 720 = 2^4 \times 3^2 \times 5^1 \]

**Factorization of 896:**
1. Divide by 2: \( 896 \div 2 = 448 \)
2. Divide by 2: \( 448 \div 2 = 224 \)
3. Divide by 2: \( 224 \div 2 = 112 \)
4. Divide by 2: \( 112 \div 2 = 56 \)
5. Divide by 2: \( 56 \div 2 = 28 \)
6. Divide by 2: \( 28 \div 2 = 14 \)
7. Divide by 2: \( 14 \div 2 = 7 \)
8. Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 896 is:
\[ 896 = 2^7 \times 7^1 \]

### Step 4: Find the HCF (Highest Common Factor)
To find the HCF, we take the lowest power of all prime factors common to both factorizations.

- For the prime factor 2: The minimum power is \( 2^4 \) (from 720).
- For the prime factor 3: It is not present in 896.
- For the prime factor 5: It is not present in 896.
- For the prime factor 7: It is not present in 720.

Thus, the only common factor is \( 2^4 \).

### Step 5: Calculate the HCF
Calculating \( 2^4 \):
\[ 2^4 = 16 \]

### Conclusion
Therefore, the greatest number that will divide 729 and 901 leaving remainders 9 and 5 respectively is:
\[ \boxed{16} \]

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