To solve the problem, we will break it down into four parts as specified in the question. We start with the number \( n = 10800 \).
### Step 1: Prime Factorization of \( n \)
First, we need to find the prime factorization of \( n \).
\[
n = 10800
\]
To factor \( 10800 \), we can divide it by prime numbers:
1. Divide by \( 2 \):
\[
10800 \div 2 = 5400
\]
\[
5400 \div 2 = 2700
\]
\[
2700 \div 2 = 1350
\]
\[
1350 \div 2 = 675 \quad (\text{stop dividing by } 2 \text{ as } 675 \text{ is odd})
\]
2. Divide by \( 3 \):
\[
675 \div 3 = 225
\]
\[
225 \div 3 = 75
\]
\[
75 \div 3 = 25 \quad (\text{stop dividing by } 3 \text{ as } 25 \text{ is not divisible by } 3)
\]
3. Divide by \( 5 \):
\[
25 \div 5 = 5
\]
\[
5 \div 5 = 1
\]
Thus, the prime factorization of \( 10800 \) is:
\[
n = 2^4 \times 3^3 \times 5^2
\]
### Step 2: Total Number of Divisors
To find the total number of divisors, we use the formula:
\[
\text{Total Divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)
\]
where \( e_1, e_2, e_3 \) are the powers of the prime factors.
For \( n = 2^4 \times 3^3 \times 5^2 \):
- \( e_1 = 4 \)
- \( e_2 = 3 \)
- \( e_3 = 2 \)
So, the total number of divisors is:
\[
(4 + 1)(3 + 1)(2 + 1) = 5 \times 4 \times 3 = 60
\]
### Step 3: Number of Even Divisors
Even divisors must include at least one factor of \( 2 \). Thus, we set \( a \) (the exponent of \( 2 \)) to be at least \( 1 \).
The possible values for \( a \) are \( 1, 2, 3, 4 \) (4 choices), and the values for \( b \) and \( c \) remain the same:
- \( b \) can be \( 0, 1, 2, 3 \) (4 choices)
- \( c \) can be \( 0, 1, 2 \) (3 choices)
So, the number of even divisors is:
\[
(4)(4)(3) = 48
\]
### Step 4: Number of Divisors of the Form \( 4m + 2 \)
Divisors of the form \( 4m + 2 \) must have \( a = 1 \) (exactly one factor of \( 2 \)).
Thus:
- \( a = 1 \) (1 choice)
- \( b \) can be \( 0, 1, 2, 3 \) (4 choices)
- \( c \) can be \( 0, 1, 2 \) (3 choices)
So, the number of divisors of the form \( 4m + 2 \) is:
\[
(1)(4)(3) = 12
\]
### Step 5: Number of Divisors that are Multiples of 15
For a divisor to be a multiple of \( 15 \), it must have at least \( 1 \) factor of \( 3 \) and \( 1 \) factor of \( 5 \).
Thus:
- \( b \) must be at least \( 1 \) (choices: \( 1, 2, 3 \) - 3 choices)
- \( c \) must be at least \( 1 \) (choices: \( 1, 2 \) - 2 choices)
- \( a \) can be \( 0, 1, 2, 3, 4 \) (5 choices)
So, the number of divisors that are multiples of \( 15 \) is:
\[
(5)(3)(2) = 30
\]
### 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**
