MAT
138-16-FALL-WEEK8-Nov.11-H1.pdf

138-16-FALL-WEEK8-Nov.11-H1.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #8 (NOVEMBER 11) Part 2. More basics on “Number Theory”. - Topic: Greatest Common Divisor. - Definition. ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , we say that 𝑑𝑑 is a “common divisor” of 𝑎𝑎 and 𝑏𝑏 if 𝑑𝑑 | 𝑎𝑎 and 𝑑𝑑 | 𝑏𝑏 . Recall: ∀𝑛𝑛 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , 𝑑𝑑 | 𝑛𝑛 ⇔ ( ∃𝑞𝑞 ∈ ℤ , 𝑛𝑛 = 𝑞𝑞𝑑𝑑 ) . Exercise: Prove or disprove each of the following statements: a) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , ( 𝑑𝑑 is a common divisor of 𝑎𝑎 and 𝑏𝑏 ) ⇒ �𝑑𝑑 |( 𝑎𝑎 + 𝑏𝑏 ) , b) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , ( 𝑑𝑑 is a common divisor of 𝑎𝑎 and 𝑏𝑏 ) ⇒ �𝑑𝑑 |( 𝑎𝑎𝑏𝑏 ) , c) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , �𝑑𝑑 |( 𝑎𝑎 + 𝑏𝑏 ) � ⇒ ( 𝑑𝑑 is a common divisor of 𝑎𝑎 and 𝑏𝑏 ) , d) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , �𝑑𝑑 |( 𝑎𝑎𝑏𝑏 ) � ⇒ ( 𝑑𝑑 is a common divisor of 𝑎𝑎 and 𝑏𝑏 ) , e) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , ( 𝑑𝑑 is a common divisor of 𝑎𝑎 and 𝑏𝑏 ) ⇒ �𝑑𝑑 ≤ max({ 𝑎𝑎 , 𝑏𝑏 }) , f) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , (( 𝑎𝑎 ≠ 0 ∨ 𝑏𝑏 ≠ 0) ( 𝑑𝑑 common divisor of 𝑎𝑎 and 𝑏𝑏 )) ⇒ �𝑑𝑑 ≤ max({| 𝑎𝑎 |, | 𝑏𝑏 |}) g) ∀𝑎𝑎 , 𝑏𝑏 , , 𝑘𝑘 ∈ ℤ , ∀𝑑𝑑 ∈ ℤ + , ( 𝑑𝑑 is a common divisor of 𝑎𝑎 and 𝑏𝑏 ) ⇒ �𝑑𝑑 |(
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