Divisibility rules (Properties of Divisibility)

this presentation is about the properties and theorems revolving Divisibility Rules.. not just "which number is divisible by this number blah blah blah.." this isnt your ordinary divisibility..

1. Divisibility Rules A lesson in Abstract Algebra Presented to Prof Jose Binaluyo

2. Divisibility Rules This presentation aims to: Define and illustrate Divisibility; and Prove statements and theorems on Divisibility tsukiscloud9

3. Definition for two given integers a and b , there exists an integer q such that b= aq , then b is divisible by a. tsukiscloud9

4. Illustrating Divisibility We use the bar “|” Example: “2|6” The notation 2|6 is read as “ 6 is divisible by 2 ” tsukiscloud9

5. Illustrating Divisibility But if a number is not divisible by another number we write “ a|b ” It’s read as “ b is not divisible to a ” tsukiscloud9

6. Consequences of the Definition a|0 or aq =0 where q=0 1|b where q=b a|a or aq =a where q=1 a | - a where q= - 1 a |±1 iff a=±1 tsukiscloud9

7. Corollary ( 2.6.1 ) tsukiscloud9 The notations a|b may also apply to negative integers a and b wherein q is a negative integer, or when a and b are both negative

8. Divisibility Theorem ( 2.6.1 ) For any integers a , b and c and a|b , then a|bc Sample: if 4|16 , then 4|96 , where c=6 tsukiscloud9

9. Divisibility Theorem ( 2.6.2 ) If a|b and b|c , then a|c Sample: if 2|4 and 4|16 , then 2|16 tsukiscloud9

10. Divisibility Theorem ( 2.6.3 ) If a|b and c|d , then ac|bd Sample: if 2|4 and 3|6 , then 2(3)|4(6) = 6|24 tsukiscloud9

11. Divisibility Theorem ( 2.6.4 ) If a|b for any integers a and b then |a|≤|b| Sample: if 2| - 4 and |2| ≤ | - 4| tsukiscloud9

12. Corollary ( 2.6.2 ) F or any integers a and b , and a|b and b|a , then a=±b Sample: if 2 | - 2 and - 2|2 then, 2=±2 tsukiscloud9

13. Divisibility Theorem ( 2.6.5 ) F or any integers a , b , and c and if a| b and a|c , then a|bx + cy. tsukiscloud9

14. Divisibility Theorem ( 2.6.5 ) Sample: if 2|4 and 2|6 | then 2|4(3)+6(2) = 2|24 , where x=3 and y=2 tsukiscloud9

15. Corollary ( 2.6.3 ) If a|b and a|c then a|b+c Sample: 3 |6 and 3|18 then, 3|6+18 = 3|24 tsukiscloud9

16. tsukiscloud9

17. Presented by yours truly, Jessa Mae Nercua Hersheys Azures BS Math CS III-E2

18. Thank You for Watching!! We hope you learned. tsukiscloud9