divisibility rules (properties of divisibility)

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.