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# Lecture_19 - MA1100 Lecture 19 Functions/Number Theory...

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1 MA1100 Lecture 19 Functions/Number Theory Inverse functions Greatest Common Divisor Euclidean Algorithm

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MA1100 Lecture 19 2 Common divisor Recall 1 Recall 2 Let a, b be integers and d a nonzero integer. Let n be an integer and m a non-zero integer. We say m divides n and write m | n if there exists an integer q such that n = mq . We also say m is a divisor of n. We say d is a common divisor (factor) of a and b if d | a and d | b .
MA1100 Lecture 19 3 Common divisor Example Let a = 48 and b = 84. Divisors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Divisors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 Common divisors of 48 and 84: 1, 2, 3, 4, 6, 12 The greatest common divisor of 48 and 84 is 12 .

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MA1100 Lecture 19 4 Greatest common divisor Definition Notation The largest integer that divides both a and b is called the greatest common divisor of a and b. gcd(a, b) Let a, b be integers, not both 0. Example gcd(48, 84) = 12 Remark gcd(0, 0) = ?
MA1100 Lecture 19 5 Basic properties of GCD 1.gcd(a, b) > 0 2.gcd(a, a) = |a| 3.gcd(a, 0) = |a| 4.gcd(a, b) = gcd(-a, b) = gcd(a, -b) = gcd(-a, -b) gcd(48, 48) = 48 gcd(-48, -48) = 48 gcd(48, 0) = 48 gcd(-48, 0) = 48 whether a and b are positive or negative Let a and b be integers with a 0.

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