1. COT 3100, Spring 2001 Applications of Discrete Structures
Section #1089X - MWF 4th period
Dr. Michael P. Frank Lecture #19 Wed., Feb. 21, 2001
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Lecture #19

2. Administrivia HW5, Quiz5 being returned Lecture Today: Friday:
Finish §2.3, the Integers and Division.
Basic number theory.
Friday:
Quiz on §
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Lecture #19

3. Review of §2.3 So Far a|b  “a divides b”  cZ: b=ac
“p is prime”  p>1  aN: (1<a<p  a|p)
Terms factor, divisor, multiple, composite.
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4. Fundamental Theorem of Arithmetic
Its "Prime Factorization"
Every positive integer has a unique representation as the product of a non-decreasing series of zero or more primes.
1 = (product of empty series) = 1
2 = 2 (product of series with one element)
4 = 2·2 (product of series 2,2)
2000 = 2·2·2·2·5·5·5; = 3·23·29
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5. Primes are Important
When you visit a secure web site ( , lock or key icon unbroken), the browser and web site are using RSA encryption.
This scheme uses public keys containing the product pq of two random large primes p and q which must be kept secret within your browser.
So the security of your web transactions depends critically on the fact that all known factoring algorithms are intractable! (Exc. quantum algs.)
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6. The Division “Algorithm”
Really just a theorem, not an algorithm…
Name used for historical reasons.
For any integer dividend a and divisor d>0, there is a unique quotient q and remainder r such that a=dq+r and 0r<d.
a,dZ, d>0, !q,r: 0r<d, a=dq+r.
Can find q and r by q=ad, r=aqd.
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7. Greatest Common Divisor
The greatest common divisor gcd(a,b) of integers a,b (not both 0) is the largest (most positive) integer d that is a divisor both of a and of b.
Example: gcd(24,36)=? Positive common divisors: 1,2,3,4,6,12… Greatest is 12.
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8. GCD shortcut
If the prime factorizations are written as a=p1a1 p2a2 …pnan and b=p1b1 p2b2 …pnbn, then the GCD is given by gcd(a,b) = p1min(a1,b1) p2min(a2,b2) …pnmin(an,bn).
Example:
a=84=2·2·3· = 22·31·71
b=96=2·2·2·2·2·3 = 25·31·70
gcd(84,96) = 22·31·70 = 2·2·3 = 12.
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9. Relative Primality
Integers a and b are called relatively prime iff their gcd = 1.
Example: 21 and 10 are not prime, but they are relatively prime. 21=3·7 and 10=2·5, so they have no common factors > 1, so their gcd = 1.
Integers a1,a2… are pairwise relatively prime if any two pairs ai, aj, ij, are rel. pr.
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10. Least Common Multiple
lcm(a,b) of positive integers a, b, is the smallest positive integer that is a multiple both of a and of b. E.g. lcm(6,10)=30
If the prime factorizations are written as a=p1a1 p2a2 …pnan and b=p1b1 p2b2 …pnbn, then the LCM is given by lcm(a,b) = p1max(a1,b1) p2max(a2,b2) …pnmax(an,bn).
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11. The mod operator An integer “division remainder” operator.
Let a,dZ with d>1. Then a mod d denotes the remainder r from the division “algorithm” with dividend a and divisor d; i.e. the remainder when a is divided by d. (Using e.g. long division.)
Can compute a mod d using ada/d.
In C programming language, “%” = mod.
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12. Modular Congruence Let Z+={nZ | n>0}, the positive integers.
Let a,bZ, mZ+.
Then a is congruent to b modulo m, written “ab (mod m)”, iff m|ab.
Also equivalent to (ab) mod m=0.
(Note this is a different use of “” than the meaning “is defined as” I’ve used before.)
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13. Useful Congruence Theorems
Let a,bZ, mZ+. Then ab (mod m)  kZ a=b+km.
Let a,b,c,dZ, mZ+. Then if ab (mod m) and cd (mod m), then:
a+c  b+d (mod m), and
ac  bd (mod m)
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