The greatest number which divides 134 and 167 leaving 2 as remainder in each case is

Options

A. 14
B. 17
C. 19
D. 33

Answer

To find the greatest number that divides both 134 and 167 leaving a remainder of 2 in each case, we can follow these steps:

### Step 1: Subtract the remainder from both numbers
First, we need to subtract the remainder (which is 2) from both numbers.

- For 134:
\[ 134 - 2 = 132 \]
- For 167:
\[ 167 - 2 = 165 \]

### Step 2: Find the HCF (Highest Common Factor) of the two results
Next, we need to find the HCF of 132 and 165.

#### Step 2.1: Prime factorization of 132
To find the HCF, we can start with the prime factorization of 132:
- 132 can be divided by 2:
\[ 132 \div 2 = 66 \]
- 66 can be divided by 2:
\[ 66 \div 2 = 33 \]
- 33 can be divided by 3:
\[ 33 \div 3 = 11 \]
- 11 is a prime number.

So, the prime factorization of 132 is:
\[ 132 = 2^2 \times 3 \times 11 \]

#### Step 2.2: Prime factorization of 165
Now, let's do the prime factorization of 165:
- 165 can be divided by 3:
\[ 165 \div 3 = 55 \]
- 55 can be divided by 5:
\[ 55 \div 5 = 11 \]
- 11 is a prime number.

So, the prime factorization of 165 is:
\[ 165 = 3 \times 5 \times 11 \]

### Step 3: Identify the common factors
Now, we can identify the common factors from the prime factorizations:
- From 132: \(2^2, 3, 11\)
- From 165: \(3, 5, 11\)

The common factors are \(3\) and \(11\).

### Step 4: Calculate the HCF
To find the HCF, we multiply the common factors:
\[ HCF = 3 \times 11 = 33 \]

### Conclusion
Thus, the greatest number that divides both 134 and 167 leaving a remainder of 2 is:
\[ \text{Answer: } 33 \]

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