Chapter 0, Problem 1E

For $n=5$ , 8, 12, 20, and 25, find all positive integers less than n and relatively prime to n.

To determine

To find all positive integers less than given n and relative prime to n

Answer to Problem 1E

All positive integers that are relative primes to:

1. 5 are 1, 2, 3 and 4 (less than 5)
2. 8 are1, 3,5 and 7 (less than 8)
3. 12 are 1, 5, 7 and 11 (less than 12)
4. 20 are 1, 3, 7, 9, 11, 13, 17 and 19 (less than 20)
5. 25 are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23 and 24 (less than 25)

Explanation of Solution

Given information:

  $n=5,8,12,20\text{ and 25}$

Concept Used:

Two numbers are relatively prime if they have no factors in common other than 1, thus two relatively prime numbers have a greatest common factor, or GCF, of 1.

Calculation:

In order to determine all positive integers less than nand relative prime to respective n we have to find all the integers less than n that have no common factor with n other than 1

First, let’s find all positive integers less than given 5and relative prime to 5, since number 1, 2, 3 and 4 have no common factor with 5 other than 1, thus 1, 2, 3 and 4 are relative prime to 5

Now, let’s find all positive integers less than given 8and relative prime to 8, since number 1, 3,5 and 7 have no common factor with 8 other than 1, thus 1, 3, 5 and 7 are relative prime to 8

Similarly, we find all positive integers less than given 12and relative prime to 12, since number 1, 5, 7 and 11 have no common factor with 12 other than 1, thus 1, 5, 7 and 11 are relative prime to 12

Similarly, we find all positive integers less than given 20and relative prime to 20, since number 1, 3, 7, 9, 11, 13, 17 and 19 have no common factor with 20 other than 1, thus 1, 3, 7, 9, 11, 13, 17 and 19 are relative prime to 20

Similarly, all positive integers less than given 25and relative prime to 25 can be found as, since number 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23 and 24 have no common factor with 25 other than 1, thus 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23 and 24 are relative prime to 25

Hence, we get all positive integers that are relative primes to:

1. 5 are 1, 2, 3 and 4 (less than 5)
2. 8 are1, 3,5 and 7 (less than 8)
3. 12 are 1, 5, 7 and 11 (less than 12)
4. 20 are 1, 3, 7, 9, 11, 13, 17 and 19 (less than 20)
5. 25 are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23 and 24 (less than 25)