1. Divisibility and Primes
ICS 6D
Sandy Irani

2. Evenly Divides x evenly divides y if y =m·x for some integer m
Denoted: x|y
y is an integer multiple (or just “multiple”) of x
x is a factor of y

3. Primes and Composites A number p is prime if
p is an integer and p > 1
The only (positive) factors of p are 1 and p.
If n > 1 is not prime, it is composite
n has a positive factor other than 1 or n.

4. Fundamental Theorem of Arithmetic
Every integer n > 1 can be expressed as a product of primes. Prime factorization is unique up to ordering.
Prime factorization of n usually written in non-decreasing order:
124 =

5. Fundamental Theorem of Arithmetic
Prime factorization of 360:

6. Exponential Notation for Prime Factorization
Write each distinct prime in the prime factorization in increasing order.
The multiplicity of a prime (# of times it appears) is written in the exponent.
124 =
360 =

7. Prime Factorization and Divisibility
If you have the prime factorization for x and y, can easily determine if x divides y:
Example: does 90 divide 1260?
1260 = 22 · 32 · 5 · 7
90 = 2 · 32 · 5

8. Prime Factorization and Divisibility
1260 = 22 · 32 · 5 · 7
40 = 23· 5
22 = 2 · 11

9. LCM and GCD Two positive integers, x and y.
The least common multiple of x and y, lcm(x,y), is the smallest integer that is a multiple of x and a multiple of y.
The greatest common divisor of x and y, gcd(x, y) is the largest integer that divides both x and y.
x and y are relatively prime if gcd(x, y) = 1

10. Prime Factorization and LCM
If you have the prime factorization for x and y, can easily determine if lcm(x, y):
44 = 22 · = 2 · 32 · 5

11. Prime Factorization and GDC
If you have the prime factorization for x and y, can easily determine if gcd(x, y):
1320 = 23 ·3 · 5 · = 22 · 32 · 5

12. Factoring: Brute-force algorithm
Input: integer N > 1
Output: (x,y) such that x > 1, y > 1 and xy = N
or “Prime” if N is prime
For x = 2 to N-1
If x evenly divides N
Return(x, N/x)
Return(“Prime”)

13. Factoring: slightly better algorithm
Input: integer N > 1
Output: (x,y) such that x > 1, y > 1 and xy = N
or “Prime” if N is prime
For x = 2 to N-1 𝑁
If x evenly divides N
Return(x, N/x)
Return(“Prime”)
If N is composite than it as
at least one factor that is at
most 𝑵

14. Factoring: slightly better algorithm
Input: integer N > 1
Output: (x,y) such that x > 1, y > 1 and xy = N
or “Prime” if N is prime
For x = 2 to N-1 𝑁
If x evenly divides N
Return(x, N/x)
Return(“Prime”)
If N has 200 digits
𝑵 has 100 digits
If inner loop takes .1 ns
Algorithm take 3 x 1082 years
Time proportional to the value
of N, not the number of digits

15. Factoring and Primality Testing
Input: integer N > 1
Output:
If N is prime:
“Prime”
If N is composite:
x > 1, y > 1, xy = N
Primality Testing
Input: integer N > 1
Output:
If N is prime:
“Prime”
If N is composite:
“Composite”
Hard: no known algorithm that
Runs in time (# digits)d
Easy: can be solved in time
(# digits)2 (probabilistic)

16. Finding Prime Numbers “Find a 200-digit prime number” How to do this?
Repeat:
Pick a random 200 digit number N
Apply primality test to N
If N is prime, return(N)
If N is composite, continue.
The number of trials until
success depends on the
density of prime numbers
among all 200 digit numbers

17. Density of primes: a first step
Theorem: there are an infinite number of prime numbers.
Euclid ~300 B.C.

18. The Prime Number Theorem
Theorem: let π(x) be the number of prime numbers in the range [2,…,x], then
lim 𝑥→∞ 𝜋(𝑥) 𝑥/( ln 𝑥) = 1
𝜋(𝑥) 𝑥 ln⁡(𝑥)

19. The Prime Number Theorem
Theorem: let π(x) be the number of prime numbers in the range [2,…,x], then
lim 𝑥→∞ 𝜋(𝑥) 𝑥/( ln 𝑥) = 1
Interpretation: A randomly chosen number in the range 2,..,x is prime with probability ~ 1 𝑙𝑛(𝑥)
To get a 200 digit prime number,
we need the density of primes in
the range from to
ln(x) ~ 2.3·(# digits in x)