Which is the largest number that will divide 1992 and 233 leaving remainders 12 and 13, respectively?

Options

A. 242
B. 220
C. 246
D. 186

Answer

To solve the problem of finding the largest number that will divide 1992 and 233, leaving remainders of 12 and 13 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
First, we need to subtract the remainders from the original numbers.

- For 1992, subtract the remainder 12:
\[ 1992 - 12 = 1980 \]

- For 233, subtract the remainder 13:
\[ 233 - 13 = 220 \]

### Step 2: Find the highest common factor (HCF)
Next, we need to find the highest common factor (HCF) of the two adjusted numbers, 1980 and 220.

### Step 3: Factorize the numbers
We will factorize both numbers.

- **Factorization of 220**:
\[ 220 = 2 \times 2 \times 5 \times 11 = 2^2 \times 5 \times 11 \]

- **Factorization of 1980**:
\[ 1980 = 2 \times 2 \times 5 \times 9 \times 11 = 2^2 \times 5 \times 11 \times 9 \]
Here, 9 can be further factored as \(3^2\), but we will focus on the common factors for HCF.

### Step 4: Identify the common factors
Now, we identify the common factors from both factorizations:

- From 220: \(2^2 \times 5 \times 11\)
- From 1980: \(2^2 \times 5 \times 11 \times 3^2\)

The common factors are \(2^2\), \(5\), and \(11\).

### Step 5: Calculate the HCF
To find the HCF, we multiply the common factors together:
\[ HCF = 2^2 \times 5 \times 11 = 4 \times 5 \times 11 = 220 \]

### Conclusion
Thus, the largest number that will divide both 1992 and 233, leaving remainders of 12 and 13 respectively, is:
\[ \boxed{220} \]