Date post: | 13-Jun-2015 |
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Education |
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Divisibility Rules
A lesson in Abstract Algebra
Presented to Prof Jose Binaluyo
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Divisibility Rules
This presentation aims to:
Define and illustrate Divisibility; and
Prove statements and theorems on Divisibility
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Definition
for two given integers a and b, there exists an integer q such that b=aq, then b is divisible by a.
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Illustrating Divisibility
We use the bar “|”
Example: “2|6”
The notation 2|6 is read as “6 is divisible by 2”
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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”
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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
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Corollary (2.6.1)
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
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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
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Divisibility Theorem (2.6.2)
If a|b and b|c, then a|c
Sample: if 2|4 and 4|16, then 2|16
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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
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Divisibility Theorem (2.6.4)
If a|b for any integers a and b then |a|≤|b|
Sample: if 2|-4 and |2| ≤|-4|
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Corollary (2.6.2)
For any integers a and b , and a|b and b|a, then a=±b
Sample: if 2|-2 and -2|2 then, 2=±2
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Divisibility Theorem (2.6.5)
For any integers a, b, and c and if a| b and a|c, then a|bx + cy.
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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
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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
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Presented by yours truly,
Jessa Mae Nercua Hersheys Azures
BS Math
CSIII-E2
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Thank You for Watching!!
We hope you learned.