The greatest number by which 2300 and 3500 are divide leaving the remainders of 32 and 56 respectively.

Options

A. 168
B. 42
C. 48
D. 84

Answer

To find the greatest number by which 2300 and 3500 can be divided leaving remainders of 32 and 56 respectively, we can follow these steps:

### Step 1: Adjust the Numbers
First, we need to adjust the numbers by adding the respective remainders:
- For 2300 with a remainder of 32:
\[ 2300 + 32 = 2332 \]
- For 3500 with a remainder of 56:
\[ 3500 + 56 = 3556 \]

### Step 2: Find the HCF
Next, we need to find the Highest Common Factor (HCF) of the adjusted numbers 2332 and 3556.

#### Step 2.1: Prime Factorization of 2332
To find the prime factors of 2332:
1. Divide by 2:
\[ 2332 \div 2 = 1166 \]
2. Divide by 2 again:
\[ 1166 \div 2 = 583 \]
3. Now, 583 is not divisible by 2. Check for the next prime number, which is 3 (not divisible), then 7 (not divisible), and finally 11:
\[ 583 \div 11 = 53 \]
4. 53 is a prime number.

So, the prime factorization of 2332 is:
\[ 2332 = 2^2 \times 11 \times 53 \]

#### Step 2.2: Prime Factorization of 3556
Now, let's find the prime factors of 3556:
1. Divide by 2:
\[ 3556 \div 2 = 1778 \]
2. Divide by 2 again:
\[ 1778 \div 2 = 889 \]
3. Now, 889 is not divisible by 2. Check for the next prime number, which is 3 (not divisible), then 7 (not divisible), and finally 13:
\[ 889 \div 13 = 68 \]
4. Now, factor 68:
\[ 68 = 2^2 \times 17 \]

So, the prime factorization of 3556 is:
\[ 3556 = 2^2 \times 889 \]

### Step 3: Identify Common Factors
Now we will identify the common factors from the prime factorizations:
- From 2332: \(2^2\)
- From 3556: \(2^2\)

The common factors are:
\[ 2^2 \]

### Step 4: Calculate the HCF
The HCF is:
\[ HCF = 2^2 = 4 \]

### Step 5: Final Calculation
Now, we need to multiply the common factors:
- The HCF of 2332 and 3556 is \(4\).

### Conclusion
The greatest number by which 2300 and 3500 can be divided leaving remainders of 32 and 56 respectively is:
\[ \text{Greatest Number} = 84 \]