What will be the greatest number that divides 1023 and 750 leaving remainders 3 and 2, respectively?

Options

A. 68
B. 65
C. 78
D. 19

Answer

To find the greatest number that divides 1023 and 750 leaving remainders of 3 and 2 respectively, we can follow these steps:

### Step 1: Adjust the numbers based on the remainders
We start by adjusting the two numbers based on the remainders given in the problem.

- For 1023, since it leaves a remainder of 3, we subtract 3:
\[ 1023 - 3 = 1020 \]

- For 750, since it leaves a remainder of 2, we subtract 2:
\[ 750 - 2 = 748 \]

### Step 2: Find the HCF of the adjusted numbers
Now we need to find the highest common factor (HCF) of the two adjusted numbers, 1020 and 748.

#### Step 2.1: Prime factorization of 1020
First, let's factor 1020:
\[ 1020 = 2 \times 510 \]
\[ 510 = 2 \times 255 \]
\[ 255 = 3 \times 85 \]
\[ 85 = 5 \times 17 \]
So, the prime factorization of 1020 is:
\[ 1020 = 2^2 \times 3^1 \times 5^1 \times 17^1 \]

#### Step 2.2: Prime factorization of 748
Next, we factor 748:
\[ 748 = 2 \times 374 \]
\[ 374 = 2 \times 187 \]
\[ 187 = 11 \times 17 \]
So, the prime factorization of 748 is:
\[ 748 = 2^2 \times 11^1 \times 17^1 \]

### Step 3: Identify common factors
Now, we identify the common prime factors from both numbers:
- The common prime factors are \(2^2\) and \(17^1\).

### Step 4: Calculate the HCF
To find the HCF, we take the lowest power of each common prime factor:
\[ HCF = 2^2 \times 17^1 = 4 \times 17 = 68 \]

### Conclusion
Thus, the greatest number that divides both 1023 and 750 leaving remainders of 3 and 2 respectively is:
\[ \boxed{68} \]