To solve the problem of finding the difference between the sum of factors of 20 and the sum of co-prime numbers of 20 up to 20, we can follow these steps:
### Step 1: Find the Factors of 20
The factors of a number are the integers that divide that number without leaving a remainder.
- The factors of 20 are: 1, 2, 4, 5, 10, and 20.
### Step 2: Calculate the Sum of the Factors of 20
Now, we need to sum these factors.
- Sum = 1 + 2 + 4 + 5 + 10 + 20 = 42
### Step 3: Identify Co-prime Numbers of 20 up to 20
Co-prime numbers are numbers that do not share any common factors with the given number (other than 1).
- The numbers from 1 to 20 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.
- Now, we check each number:
- 1 (co-prime)
- 2 (not co-prime, common factor 2)
- 3 (co-prime)
- 4 (not co-prime, common factor 4)
- 5 (not co-prime, common factor 5)
- 6 (not co-prime, common factor 2)
- 7 (co-prime)
- 8 (not co-prime, common factor 4)
- 9 (co-prime)
- 10 (not co-prime, common factor 10)
- 11 (co-prime)
- 12 (not co-prime, common factor 4)
- 13 (co-prime)
- 14 (not co-prime, common factor 2)
- 15 (not co-prime, common factor 5)
- 16 (not co-prime, common factor 4)
- 17 (co-prime)
- 18 (not co-prime, common factor 2)
- 19 (co-prime)
- 20 (not co-prime, common factor 20)
- The co-prime numbers of 20 up to 20 are: 1, 3, 7, 9, 11, 13, 17, and 19.
### Step 4: Calculate the Sum of Co-prime Numbers
Now, we sum these co-prime numbers.
- Sum = 1 + 3 + 7 + 9 + 11 + 13 + 17 + 19
- Breaking it down:
- 1 + 3 = 4
- 4 + 7 = 11
- 11 + 9 = 20
- 20 + 11 = 31
- 31 + 13 = 44
- 44 + 17 = 61
- 61 + 19 = 80
### Step 5: Find the Difference
Finally, we find the difference between the sum of the factors of 20 and the sum of the co-prime numbers.
- Difference = Sum of Co-prime Numbers - Sum of Factors
- Difference = 80 - 42 = 38
### Final Answer
The difference between the sum of the factors of 20 and the sum of co-prime numbers of 20 up to 20 is **38**.
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