To solve the question of finding the difference between the number of even divisors and the number of prime divisors of 630, we can follow these steps:
### Step 1: Find the prime factorization of 630
To find the prime factorization of 630, we can divide it by the smallest prime numbers:
- 630 ÷ 2 = 315
- 315 ÷ 3 = 105
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 7 is a prime number.
Thus, the prime factorization of 630 is:
\[ 630 = 2^1 \times 3^2 \times 5^1 \times 7^1 \]
### Step 2: Count the number of even divisors
Even divisors are those divisors that include at least one factor of 2. To find the total number of divisors, we can use the formula:
\[
\text{Total Divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1)
\]
where \( e_1, e_2, e_3, \) and \( e_4 \) are the exponents in the prime factorization.
For 630:
- The exponents are 1 (for 2), 2 (for 3), 1 (for 5), and 1 (for 7).
- So, the total number of divisors is:
\[
(1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 \times 3 \times 2 \times 2 = 24
\]
To find the number of even divisors, we can subtract the number of odd divisors (which do not include the factor of 2):
- Odd divisors are calculated using the factors without 2:
\[
(2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 = 12
\]
Thus, the number of even divisors is:
\[
\text{Even Divisors} = \text{Total Divisors} - \text{Odd Divisors} = 24 - 12 = 12
\]
### Step 3: Count the number of prime divisors
From the prime factorization \( 630 = 2^1 \times 3^2 \times 5^1 \times 7^1 \), we can see that the prime divisors are 2, 3, 5, and 7. Therefore, the number of prime divisors is:
- Prime Divisors = 4 (2, 3, 5, 7)
### Step 4: Calculate the difference
Now, we find the difference between the number of even divisors and the number of prime divisors:
\[
\text{Difference} = \text{Even Divisors} - \text{Prime Divisors} = 12 - 4 = 8
\]
### Final Answer
The difference between the number of even divisors and the number of prime divisors of 630 is **8**.
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