To find all the factors of 90, we can follow these steps:
### Step 1: Prime Factorization of 90
First, we need to factor 90 into its prime components.
- Start by dividing 90 by the smallest prime number, which is 2:
\[
90 \div 2 = 45
\]
- Next, we factor 45 by the next smallest prime number, which is 3:
\[
45 \div 3 = 15
\]
- We continue factoring 15 by 3:
\[
15 \div 3 = 5
\]
- Finally, we have 5, which is a prime number.
Thus, the prime factorization of 90 is:
\[
90 = 2^1 \times 3^2 \times 5^1
\]
### Step 2: Use the Formula to Find the Total Number of Factors
To find the total number of factors, we use the formula:
\[
\text{Total Factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)
\]
where \(e_1, e_2, e_3\) are the powers of the prime factors.
For 90:
- The power of 2 is 1, so \(e_1 = 1\)
- The power of 3 is 2, so \(e_2 = 2\)
- The power of 5 is 1, so \(e_3 = 1\)
Now, we calculate:
\[
\text{Total Factors} = (1 + 1)(2 + 1)(1 + 1) = 2 \times 3 \times 2 = 12
\]
### Step 3: List All the Factors
Now that we know there are 12 factors, we can find them by checking the divisibility of numbers up to 90:
1. **1** (90 ÷ 1 = 90)
2. **2** (90 ÷ 2 = 45)
3. **3** (90 ÷ 3 = 30)
4. **5** (90 ÷ 5 = 18)
5. **6** (90 ÷ 6 = 15)
6. **9** (90 ÷ 9 = 10)
7. **10** (90 ÷ 10 = 9)
8. **15** (90 ÷ 15 = 6)
9. **18** (90 ÷ 18 = 5)
10. **30** (90 ÷ 30 = 3)
11. **45** (90 ÷ 45 = 2)
12. **90** (90 ÷ 90 = 1)
### Conclusion
The complete list of factors of 90 is:
\[
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
\]
