4383_Notes_2o3 - Math 4383 Section 2.3 Page 1 of 7 Number...

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Unformatted text preview: Math 4383 Section 2.3 Page 1 of 7 Number Theory Chapter 2 Section 2.3 – Greatest Common Factor Language for Division 3 is a factor of 12 3 is a divisor of 12 12 is a multiple of 3, 12 is a multiple of 4 12 is divisible by 3, 12 is divisible by 4 Formal Language – this is IMPORTANT An integer b is said to be divisible by an integer ≠ a , in symbols | a b , if there exists an integer c such that = b ac . We write | a b if no such integer c exists. | a b is usually read “a divides b” Examples of the notation Positive and Negative Divisors Basic Properties: 1. | 0 a , 1| a and | a a for all integers a 2. |1 a if and only if 1 = ± a 3. If | a b and | c d then | ac bd 4. If | a b and | b c , then | a c 5. | a b and | b a if and only if = ± a b 6. If | a b and ≠ b , then ≤ a b 7. If | a b and | a c , then | ( ) + a bx cy for all integers x and y Number 7 extends to more summands – Math 4383 Section 2.3 Page 2 of 7 Definition – Greatest Common Divisor Given integers a and b, not both zero, the positive integer d is called the greatest common divisor...
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4383_Notes_2o3 - Math 4383 Section 2.3 Page 1 of 7 Number...

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