Euler function
The Euler function counts the number of positive whole numbers that are relatively prime to a given number.
Imagine you have a bunch of toys that you love to play with, but you need to share them with others. The Euler function is like counting the number of toys that nobody else has, and are only yours to play with.
For example, let's say our number is 10. We start by listing out all the positive whole numbers less than or equal to 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Now, let's go through each of these numbers and see if they are relatively prime to 10, which means they have no common factors other than 1.
- 1 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 2 is relatively prime to 10 because it only has a common factor of 2 with 10.
- 3 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 4 is not relatively prime to 10 because it has a common factor of 2 with 10.
- 5 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 6 is not relatively prime to 10 because it has a common factor of 2 with 10.
- 7 is relatively prime to 10 because it has no common factors with 10.
- 8 is not relatively prime to 10 because it has a common factor of 2 with 10.
- 9 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 10 is not relatively prime to 10 because it only has a common factor of 10 with 10.
So out of the 10 numbers, only 4 of them (1, 3, 7, and 9) are relatively prime to 10. Therefore, the Euler function of 10 is 4.
Overall, the Euler function helps us count the "special" numbers that can't be divided by any other positive whole number except 1, and tells us how many of them there are for a given number.
Imagine you have a bunch of toys that you love to play with, but you need to share them with others. The Euler function is like counting the number of toys that nobody else has, and are only yours to play with.
For example, let's say our number is 10. We start by listing out all the positive whole numbers less than or equal to 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Now, let's go through each of these numbers and see if they are relatively prime to 10, which means they have no common factors other than 1.
- 1 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 2 is relatively prime to 10 because it only has a common factor of 2 with 10.
- 3 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 4 is not relatively prime to 10 because it has a common factor of 2 with 10.
- 5 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 6 is not relatively prime to 10 because it has a common factor of 2 with 10.
- 7 is relatively prime to 10 because it has no common factors with 10.
- 8 is not relatively prime to 10 because it has a common factor of 2 with 10.
- 9 is relatively prime to 10 because it only has a common factor of 1 with 10.
- 10 is not relatively prime to 10 because it only has a common factor of 10 with 10.
So out of the 10 numbers, only 4 of them (1, 3, 7, and 9) are relatively prime to 10. Therefore, the Euler function of 10 is 4.
Overall, the Euler function helps us count the "special" numbers that can't be divided by any other positive whole number except 1, and tells us how many of them there are for a given number.