week04 - Congruences(Part 2 Fundamental Theorem of...

Congruences (Part 2) & Fundamental Theorem of Arithmetic
Original Notes adopted fromOctober 2, 2001 (Week 4)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
Fundamental Theorem of Arithmetic
Every natural number other than 1 can be represented as a product of primes and that representation is
unique up to the order of factors.
Eg. = 5 * 30 = 5 * 3 * 10 = 5 * 3* 5* 2
= 2 * 3* 5* 5
Given n, write
n = p
1
α 1
, p
2
α 2
… p
α k
where each p
i
is prime and each α
i
is a natural number.
Eg. 7
249
≠ 3
568
Eg. 12|36 (12 is not prime) 36 = 4 * 9, 12 doesn’t divide into 4, 12 doesn’t divide into 9.
Theorem : If p is prime & a, b ∈ N, and if p | (ab) then p|a or p|b
Proof: If a or b = 1, done.
If not a = p
1
α 1
….p
s
α s
b = q
1
β 1
… q
t
β t
ab = p
1
α 1
…. p
s
α s
q
1
β 1
….q
t
β t
Given ab = pk, some k
∴ p occurs in factorization of ab into primes
p = p
i
some i, or p = q
j
, some j
If p = p
i
, then p|a
If p = q
j
, then p |a
Recall for m > 1 a A ± b (mod m) if m | a-b
Eg. m = 5 (remainders of 0,1,2,34)
Any a is congruent to 1 of {0,1,2,3,4} mod 5.