The number of even divisors of 100 will be

Options

A. 7
B. 6
C. 5
D. 8

Answer

To find the number of even divisors of 100, we can follow these steps:

### Step 1: Prime Factorization of 100
First, we need to find the prime factorization of 100.
- 100 can be expressed as \( 10 \times 10 \).
- Each 10 can be expressed as \( 2 \times 5 \).
- Therefore, \( 100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2 \).

### Step 2: Identify Even Divisors
To find the even divisors, we need to consider the prime factorization we found. An even divisor must include at least one factor of 2.

### Step 3: Calculate Total Divisors
The formula for finding the total number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is:
\[ (e_1 + 1)(e_2 + 1) \]
For \( 100 = 2^2 \times 5^2 \):
- The exponents are \( e_1 = 2 \) (for 2) and \( e_2 = 2 \) (for 5).
- Therefore, the total number of divisors is \( (2 + 1)(2 + 1) = 3 \times 3 = 9 \).

### Step 4: Calculate Odd Divisors
Next, we need to find the number of odd divisors, which do not include the factor of 2. The odd divisors come only from the factor of 5.
- The prime factorization for odd divisors is \( 5^2 \).
- The number of odd divisors is \( (2 + 1) = 3 \).

### Step 5: Calculate Even Divisors
To find the number of even divisors, we subtract the number of odd divisors from the total number of divisors:
\[ \text{Even Divisors} = \text{Total Divisors} - \text{Odd Divisors} = 9 - 3 = 6 \]

### Final Answer
Thus, the number of even divisors of 100 is **6**.

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