Greatest Common Divisor GCD Programming

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Greatest Common Divisor (GCD) Programming

• Jonas Tan
• jonastan_at_cs.tamu.edu

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Common Divisor

• A common divisor is a positive integer can be a
  factor of two positive integers, a and b.
• Example
• The common divisors of 12 and 18 are 1, 2, 3, 6.

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Greatest Common Divisor

• The GCD of two integers, a and b, is the largest
  positive integer that divides both a and b.
• Example
• gcd(12, 18) 6
• gcd(-15, 5) 5

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Euclidean Algorithm

• Euclidean algorithm is an efficient and simple
  way to calculate GCD.
• Theorem
• Let a and b be positive integers, and r the
  remainder. When a is divided by b. Then (a, b)
  (b, r).

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Proof of Euclidean Algorithm

• Let d (a, b) and d (b, r). To prove d d,
  it suffices to show that dd and dd.
• a bq r (1)
• Show dd
• d (a, b) gt da and db gt d(a-bq).
• In other words, dr by (1).
• Thus, db and dr, so d(b,r) that is, dd.
• Show dd
• d (b, r) gt db and dr gt d(bqr).
• In other words, da by (1).
• Thus, db and da, so d(a,b) that is, dd.
• gt d d, that is, (a, b) (b, r)

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Euclidean Algorithm
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Example

• gcd(68, 26)
• gcd(16, 10)
• gcd(10, 6)
• gcd(6, 4)
• gcd(4, 2)
• gcd(2, 0)
• Therefore, gcd(68, 26) 2

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Program Exercise

• Open \2000SummerProject\NumberTheoryUI\2006Main.sl
  n
• Open project Gcd
• Add your codes into gcd.cs
• BigInteger gcd(BigInteger x, BigInteger y)
• //Euclidean Algorithm