To solve the problem step by step, we will first find the prime factorization of \( n = 10800 \) and then use it to find the required quantities.
### Step 1: Prime Factorization of \( n = 10800 \)
To find the prime factorization, we can divide \( 10800 \) by prime numbers:
1. Divide by \( 2 \):
- \( 10800 \div 2 = 5400 \)
- \( 5400 \div 2 = 2700 \)
- \( 2700 \div 2 = 1350 \)
- \( 1350 \div 2 = 675 \) (stop here as \( 675 \) is odd)
2. Divide by \( 3 \):
- \( 675 \div 3 = 225 \)
- \( 225 \div 3 = 75 \)
- \( 75 \div 3 = 25 \) (stop here as \( 25 \) is not divisible by \( 3 \))
3. Divide by \( 5 \):
- \( 25 \div 5 = 5 \)
- \( 5 \div 5 = 1 \)
Thus, the prime factorization of \( 10800 \) is:
\[
10800 = 2^4 \times 3^3 \times 5^2
\]
### Step 2: Total Number of Divisors
The formula for finding the total number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \) is:
\[
(e_1 + 1)(e_2 + 1)(e_3 + 1)
\]
For \( 10800 = 2^4 \times 3^3 \times 5^2 \):
- \( e_1 = 4 \) (for \( 2 \))
- \( e_2 = 3 \) (for \( 3 \))
- \( e_3 = 2 \) (for \( 5 \))
Calculating the total number of divisors:
\[
(4 + 1)(3 + 1)(2 + 1) = 5 \times 4 \times 3 = 60
\]
### Step 3: Number of Even Divisors
An even divisor must have at least one factor of \( 2 \). Therefore, we can consider the powers of \( 2 \) from \( 1 \) to \( 4 \):
- Possible values for \( a \) (power of \( 2 \)): \( 1, 2, 3, 4 \) (4 choices)
- Possible values for \( b \) (power of \( 3 \)): \( 0, 1, 2, 3 \) (4 choices)
- Possible values for \( c \) (power of \( 5 \)): \( 0, 1, 2 \) (3 choices)
Calculating the number of even divisors:
\[
4 \times 4 \times 3 = 48
\]
### Step 4: Number of Divisors of the Form \( 4m + 2 \)
For a divisor to be of the form \( 4m + 2 \), it must have exactly one factor of \( 2 \):
- \( a = 1 \) (1 choice)
- Possible values for \( b \) (power of \( 3 \)): \( 0, 1, 2, 3 \) (4 choices)
- Possible values for \( c \) (power of \( 5 \)): \( 0, 1, 2 \) (3 choices)
Calculating the number of divisors of the form \( 4m + 2 \):
\[
1 \times 4 \times 3 = 12
\]
### Step 5: Number of Divisors that are Multiples of 15
A divisor is a multiple of \( 15 \) if it includes at least one factor of \( 3 \) and one factor of \( 5 \):
- Possible values for \( a \) (power of \( 2 \)): \( 0, 1, 2, 3, 4 \) (5 choices)
- Possible values for \( b \) (power of \( 3 \)): \( 1, 2, 3 \) (3 choices)
- Possible values for \( c \) (power of \( 5 \)): \( 1, 2 \) (2 choices)
Calculating the number of divisors that are multiples of \( 15 \):
\[
5 \times 3 \times 2 = 30
\]
### Final Answers
a. Total number of divisors of \( n \) = **60**
b. Number of even divisors = **48**
c. Number of divisors of the form \( 4m + 2 \) = **12**
d. Number of divisors which are multiples of \( 15 \) = **30**
