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. 1236
(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 pa or pb
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 pa
If p = q
j
, then p a
Recall for m > 1 a
A
±
b (mod m) if m  ab
Eg. m = 5 (remainders of 0,1,2,34)
Any a is congruent to 1 of {0,1,2,3,4} mod 5.
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 Spring '10
 Applebaugh
 Math, Number Theory, Congruence, Natural number, Prime number

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