Find the largest number which when it divides 1156 and 1231 leaves a remainder of 12 and 9 respectively.

Answer

To find the largest number that, when dividing 1156 and 1231, leaves a remainder of 12 and 9 respectively, we can follow these steps:

### Step 1: Subtract the remainders from the given numbers.
- For 1156, subtract the remainder 12:
\[ 1156 - 12 = 1144 \]
- For 1231, subtract the remainder 9:
\[ 1231 - 9 = 1222 \]

### Step 2: Find the highest common factor (HCF) of the two results obtained.
We need to find the HCF of 1144 and 1222.

### Step 3: Prime factorization of 1144.
- Divide by 2:
\[ 1144 \div 2 = 572 \]
- Divide by 2 again:
\[ 572 \div 2 = 286 \]
- Divide by 2 again:
\[ 286 \div 2 = 143 \]
- Now, divide 143 by 11:
\[ 143 \div 11 = 13 \]
- So, the prime factorization of 1144 is:
\[ 1144 = 2^3 \times 11 \times 13 \]

### Step 4: Prime factorization of 1222.
- Divide by 2:
\[ 1222 \div 2 = 611 \]
- Now, divide 611 by 13:
\[ 611 \div 13 = 47 \]
- So, the prime factorization of 1222 is:
\[ 1222 = 2^1 \times 13 \times 47 \]

### Step 5: Identify the common factors.
From the prime factorizations:
- 1144: \(2^3 \times 11 \times 13\)
- 1222: \(2^1 \times 13 \times 47\)

The common factors are:
- The minimum power of 2 is \(2^1\).
- The common factor 13 is present in both.

### Step 6: Calculate the HCF.
The HCF is obtained by multiplying the common factors:
\[ \text{HCF} = 2^1 \times 13 = 2 \times 13 = 26 \]

### Conclusion:
The largest number that divides both 1156 and 1231 leaving remainders of 12 and 9 respectively is **26**.

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