Congruences (Part 3) & Intro to Rational Numbers
Original Notes adopted fromOctober 9, 2001 (Week 5)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
a & b are
relatively prime
if their only common factor is 1
Lemma
:
If m  ab and m & a are relatively prime, then m  b
Fix m > 1
∅
(m) = number of numbers in { 1,2,..., m – 1} that are relatively prime with m.
If m is prime,
∅
(m) = p –1.
Eg. m = 6 {1,2,3,4,5}
∅
(6) = 2...
(have no common factors with 6)
Lemma: If a & and m are relatively prime & b and m are relatively prime, then ab & m are
relatively prime.
Opposite of Relatively Prime: Have a common factor that is prime.
Proof:
If not, there exists p prime such that p m and p ab
∴
pa or p b.
If p a,
a & m not relatively prime.
If p b, b & m not relatively prime
....
Contradiction.
Note: If a
≡
b (mod m ) and a & m are relatively prime, then b & m are relatively prime.
For a = b + tm; some t
∈
Z.
Euler's Theorem
: For any m > 1, if a is relatively prime to m, then a
∅
m
≡
1 (mod m).
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 Spring '10
 Applebaugh
 Math, Number Theory, Congruence, Prime number, Greatest common divisor

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