Fundamental Theorem of
Arithmetic
Euclid's Lemma
• If p is a prime that divides ab, then p
divides a or p divides b.
Examples
• 17 is a prime divisor of 6,052*9,872
Check that 17 | 59,745,344
By Euclid's Lemma 17|6052 or 17|9872.
• 6 divides 4*3 = 12
6 does not divide either 4 or 3.
How can this be?
• 6 is not prime!
Proof of Euclid's Lemma
• Suppose p does not divide a.
• Since p is prime, p and a must be relatively
prime
• So there must be integers s,t such that
1 = ps + at.
• But then b = psb + abt
• Since p divides both terms on the right, p | b.
Fundamental Theorem of
Arithmetic
• Every integer greater than 1 is a prime
or a product of primes.
This product is unique, except for the
order in which the factors appear.
• That is, if n = p1p2…pr and n = q1q2…qs,
where the p's and q's are primes, then
• r = s and, after renumbering the q's, we
have pi = qi for all i.
Sketch of Proof
• Suppose wlog, r ≤ s.
• Since p1 is prime, and p1|q1q2…qs, then
by Euclid's lemma, p1|qi for some i.
• Since qi is prime, p1 = qi.
• Renumber so that p1= q1.
• Repeat: p2 = q2 … pr=qr.
• There can be no q's left over, so s = r!