Question

Two numbers are said to be 'relatively prime' if their greatest common factor is 1. How many integers greater than 10 and less than 30 are relatively prime with 28?

Answer 1

The number of integers greater than 10 and less than 30 which are relative prime with 28 is: nine (9).

Explanation:

Two numbers are said to be relative prime if their greatest common factor is 1.

Integer greater than 10 and less than 30 are:

11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29

Let list all the factors of each number and compare individually with 28 to determine if it is relative prime with 28.

11 = 1, 11

12 = 1, 2, 3, 4, 6, 12

13 = 1, 13

14 = 1, 2, 7, 14

15 = 1, 3, 5, 15

16 = 1, 2, 4, 8, 16

17 = 1, 17

18 = 1, 2, 3, 6, 9, 18

19 = 1, 19

20 = 1, 2, 4, 5, 10, 20

21 = 1, 3, 7, 21

22 = 1, 2, 11, 22

23 = 1, 23

24 = 1, 2, 3, 4, 6, 8, 12, 24

25 = 1, 5, 25

26 = 1, 2, 13, 26

27 = 1, 3, 9, 27

28 = 1, 2, 4, 7, 14, 28

29 = 1, 29

Let remove all the numbers that have a factor common with 28 other than 1 because we only want 1 to be the common factor. We are left with the following list:

11, 13, 15, 17, 19, 23, 25, 27, 29.

If we check the greater common factor (GCF) for each of the above number relative to 28, we get 1.

1. GCF{11, 28} = 1
2. GCF{13, 28} = 1
3. GCF{15, 28} = 1
4. GCF{17, 28} = 1
5. GCF{19, 28} = 1
6. GCF{23, 28} = 1
7. GCF{25, 28} = 1
8. GCF{27, 28} = 1
9. GCF{28, 29} = 1