50 people work in a cooperative society. They all use their own conveyance. 20 people use their scooters, 12 go by their cars, 16 go by public transport and 2 use bicycle. Find H.C.F. of 20, 16, 12 and 2.

Options

A. 2
B. 4
C. 6
D. 5

Answer

To find the Highest Common Factor (H.C.F.) of the numbers 20, 12, 16, and 2, we will use the prime factorization method. Here’s the step-by-step solution:

### Step 1: Prime Factorization of Each Number
1. **Prime factorization of 20**:
- 20 can be divided by 2: \( 20 \div 2 = 10 \)
- 10 can be divided by 2: \( 10 \div 2 = 5 \)
- 5 is a prime number.
- Therefore, the prime factorization of 20 is \( 2^2 \times 5^1 \).

2. **Prime factorization of 12**:
- 12 can be divided by 2: \( 12 \div 2 = 6 \)
- 6 can be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.
- Therefore, the prime factorization of 12 is \( 2^2 \times 3^1 \).

3. **Prime factorization of 16**:
- 16 can be divided by 2: \( 16 \div 2 = 8 \)
- 8 can be divided by 2: \( 8 \div 2 = 4 \)
- 4 can be divided by 2: \( 4 \div 2 = 2 \)
- 2 is a prime number.
- Therefore, the prime factorization of 16 is \( 2^4 \).

4. **Prime factorization of 2**:
- 2 is a prime number itself.
- Therefore, the prime factorization of 2 is \( 2^1 \).

### Step 2: List the Prime Factorizations
- 20 = \( 2^2 \times 5^1 \)
- 12 = \( 2^2 \times 3^1 \)
- 16 = \( 2^4 \)
- 2 = \( 2^1 \)

### Step 3: Identify the Common Factors
Now, we identify the common prime factors from the factorizations:
- The only common prime factor among all four numbers is 2.

### Step 4: Determine the Lowest Power of the Common Factor
- The powers of 2 in the factorizations are:
- For 20: \( 2^2 \)
- For 12: \( 2^2 \)
- For 16: \( 2^4 \)
- For 2: \( 2^1 \)

The lowest power of 2 among these is \( 2^1 \).

### Step 5: Calculate the H.C.F.
Thus, the H.C.F. of 20, 12, 16, and 2 is:
\[ H.C.F. = 2^1 = 2 \]

### Final Answer
The H.C.F. of 20, 12, 16, and 2 is **2**.

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