Download AMC Number Theory Part 1

 American Math Competition Name: Introduction to Number Theory
Prime Numbers
Definition: A prime number, p, is a natural number such that if p | ab then p | a or p | b.
In other words, p is prime if every time it divides the product of two numbers, it must divide one of the
numbers. Some of you may have heard a different definition of prime.
Definition: An irreducible number, p, is a natural number such that if n | p then n  1 or n  p.
In other words, an irreducible number is a number that can only be divided by itself and 1.
It so happens, that in the natural numbers, irreducible and prime are the equivalent.
The chart above yields one method of finding prime numbers. It is called the Sieve of Eratosthenes.
1. Circle 2, because it is prime. Cross out all multiples of 2.
2. Circle 3, because it is prime. Cross out all multiple of 3.
3. Repeat Process until all numbers are marked.
** If a natural number has no prime divisors less than its square root, then it is prime.
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number Theory American Math Competition Name: Euclidean algorithm
As we noted before, for any two numbers, m and n , if n < m, then we can arrive at the greatest common
factor of n and m, gcd(n , m), by reiterating the process below
m  nq1  r1
n  r1q2  r2
r1  r2 q3  r3
...
rn  rn 1qn  2  0
Then rn 1 , the last nonzero remainder is the greatest common divisor.
This algorithm doesn’t only allow us to compute the gcd of two numbers rather quickly and efficiently,
but it also leads us to a very useful conclusion. Namely, that if
d  gcd(n, m), then there exists a, b   such that
d  an  bm
Example: In the previous set of notes, you were asked to use the Euclidean Algorithm to find
gcd(216, 52) .
216  52(4)  8
52  8(6)  4
8  4(2)
So gcd(216,52) = 4. Now we will use the Euclidean Algorithm backwards to write 4 as a linear
combination of 216 and 52.
4  52  8(6)
8  216  52(4)
4  52  -(216  -52(4))(6)
4  52  (6)216  (24)(52)
4  (25)52  (6)216
by equation 2
by equation 1
by substitution
by distribution
Exercises with the Euclidean Algorithm
Problem1. Find the greatest common divisor of 52 and 216 using the Euclidean Algorithm.
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number Theory American Math Competition Name: ** The least common multiple of m and n is the product mn divided by the gcd(m,n).
Problem 2. Find the least common multiple of 52 and 216.
Problem3. Find the smallest positive integer, N, such that the value 7  30N is not a prime number.
Problem 4. The product of a set of positive integers is 140. What is their least possible sum?
Counting Divisors
Suppose you wanted to know how many factors of 220 existed. You could just start counting, but that
would take a while.
However, we know that the prime factorization of 220 is
220  22 ·5·11
That means that you could find divisors by taking groups of divisors
{1, 2, 2·5, 2·5·11, 2·11, 22 , 22 ·5, 22 ·11, 5, 11, 5·11, 22 ·5·11} .
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number Theory American Math Competition Name: There are 12 divisors (or factors) of 220.
But this is not an efficient way of counting the number of divisors, t (n) , for a arbitrary number n.
What we can notice, is that there are 3 choices of powers of 2 because we can choose any power from
20 , 21 , 22
For each of these powers we can choose 2 powers of 5 because we can pair them up with
50 ,51
And similarly, we can join this pair with any of the two powers of 11
110 ,111
So a more efficient way of counting the number of divisors is t (220)  (2  1)(1  1)(1  1)  12
** Theorem:
Let n be a natural number with prime factorization
n  p1e1 · p2e2 · p3e3 · p4e4  pn en
be the prime factorization of n. Then the number of factors
of n is given by
t (n)  (e1  1)(e2  1) (en  1)
Example: Find the number of factors of 200.
Problem 1. Show that any positive perfect square has an odd number of positive divisors.
Is the converse also true?
Complementary Counting
This is just the principle of counting the number of things you don’t want and subtracting from the total
number.
Example: How many divisors of 5400 are not multiples of any perfect square greater than 1?
Solution:
Step 1: Prime factorization of 5400
Step 2: What should a divisor of 5400 look like, given your answer to step 1.
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number Theory American Math Competition Name: Step 3: Count the number of perfect square divisors and subtract it from the total number of divisors.
Problem 1: If n has exactly 7 positive divisors, how many positive divisors does n 2 have?
Problem 2: How many of the positive divisors of 960 have 6 positive divisors?
Problem 3: A certain integer has 20 positive divisors.
a)
b)
c)
d)
What is the smallest number of primes that could divide the integer?
What is the largest number of primes that could divide the integer?
What is the smallest natural number that has exactly 20 positive divisors?
Are there any natural numbers smaller than your answer that have more the 20 positive divisors?
Problem 4: Find the sum of the perfect square divisors of the smallest integer with exactly 6 perfect
square divisors.
Problem 5: What is the sum of all positive integers less than 100 that have exactly twelve divisors?
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number Theory