To find the factors of a number, you need to identify all the integers that divide the number exactly (i.e., without leaving a remainder). Let's break down the process step-by-step for the number 180.
### Step-by-Step Solution
1. Understand the Problem:
- We need to find all the positive integers that divide 180 exactly.
2. Initialize:
- Let's start testing from the smallest integer.
3. Testing for Factors:
- Begin with 1. Since every number is divisible by 1:
- 180 ÷ 1 = 180 → 1 is a factor.
- Move to the next integer, 2.
- 180 is an even number, so it is divisible by 2:
- 180 ÷ 2 = 90 → 2 is a factor.
- Next, test 3:
- Sum of the digits of 180 is 1 + 8 + 0 = 9, which is divisible by 3.
- 180 ÷ 3 = 60 → 3 is a factor.
- Test 4:
- 180 ÷ 4 = 45 → 4 is a factor.
- Test 5:
- Last digit of 180 is 0, which means it is divisible by 5.
- 180 ÷ 5 = 36 → 5 is a factor.
- Test 6:
- 180 ÷ 6 = 30 → 6 is a factor.
- Continue testing with the following integers:
- 180 ÷ 9 = 20 → 9 is a factor.
- 180 ÷ 10 = 18 → 10 is a factor.
- 180 ÷ 12 = 15 → 12 is a factor.
- 180 ÷ 15 = 12 → 15 is a factor.
- 180 ÷ 18 = 10 → 18 is a factor.
- 180 ÷ 20 = 9 → 20 is a factor.
- 180 ÷ 30 = 6 → 30 is a factor.
- 180 ÷ 36 = 5 → 36 is a factor.
- 180 ÷ 45 = 4 → 45 is a factor.
- 180 ÷ 60 = 3 → 60 is a factor.
- 180 ÷ 90 = 2 → 90 is a factor.
- 180 ÷ 180 = 1 → 180 is a factor.
4. List all Factors:
- Combining all the factors, we get:
[tex]\[
\{1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180\}
\][/tex]
These are all the integer factors of 180.
