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Handouts: Course Outline
Note: The following pages should be copied on one side only and
distributed to participants at the completion of each section.
Note that the Course Outline should not initially be included in the notebook, but distributed to
participants throughout the course as the appropriate sections are completed. They should initially
be withheld from the notebook since participants are led to make conjectures in order to develop
many of the ideas and theorems introduced in the course; having the outline in hand during these
discussions would give away too many “punch-lines”. However, the outline of each section
should be distributed as that section is completed so that participants have a complete list of
theorems and properties to serve as a valuable reference. It is recommended that the outline be
copied on one side only so that the appropriate page can be distributed as soon as possible at the
completion of a section.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
1
Section 1: Integers and Divisibility
Definition: divides
Theorem 1.1: If d | a and d |b then d |(a  b) . Prove.
Theorem 1.2: If a|b and b| c then a| c . Prove as in-class exercise.
Theorem 1.3: If d | a then d | ca for c  Z. Prove as in-class exercise.
Theorem 1.4: If d | a and d |b then d | c1a  c2 b for c1 , c2  Z. Prove as in-class exercise.
Theorem 1.4e: Given d | a1 , d | a 2 , … and d | a n , then d |(c1a1  c2 a 2   cn an ) for
ci  Z, i  1,2,n .
Examples:
1) Can a collection of dimes and quarters have a value of $2.06?
No: 5|(10d + 25q) but 5 does not divide 206
2) Given 100 coins consisting of pennies, dimes and quarters, can
their value total $5.00?
No: Subtract p+d+q=100 from p+10d+25q=500. The
integer 3 divides the left hand side of the resulting equation
but not the right hand side.
Definition: greatest common divisor of a and b , written ( a , b)
Examples (using notation): Find (80, 12), (121,136), (1234, 34560), (0, n)
Definition: relatively prime
Theorem 1.5: (Division Algorithm) Given integers a and b , with
b  0 , there exist unique integers q and r where 0  r  b such
that a  bq  r .
Theorem 1.6: If a  bq  r then (a , b)  (b, r ) .
Examples: Euclidean Algorithm for finding the GCD.
1) (885, 330)
4) (1234, 34560)
2) (80, 12)
5) (231, 20)
3) (121, 136)
Activity: Euclidean Algorithm, Geometrically Speaking (if time)
Key: a = 18cm, b1 = 12cm, b2 = 10cm, b3 = 5cm
Theorem 1.7: If (a , b)  d then there exist integers x and y so that ax  by  d .
Example: Find integers x, y such that 9x + 12y = 3 by inspection.
Examples: “Backwards” or Extended Euclidean Algorithm
1) 885x + 330y = 15
4) 1234x + 34560y = 2
2) 80x + 12y = 4
5) 231x + 20y = 1
3) 121x + 136y = 1
Theorem 1.8: If d | ab and (a , d )  1 then d |b . Prove.
Theorem 1.9: If a| m and b| m and (a , b)  1 then ab| m . Prove.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
2
Section 2: Primes and Factorization
Definition: prime
Theorem 2.1: Every integer n  1 is divisible by a prime.
Theorem 2.2: Every integer n  1 can be written as a product of primes. Explain.
Example: How do you determine if a number is prime?
1) Sieve of Eratosthenes (Activity)
2) Recall factoring 1234. How did you determine 617 was prime?
(Lead-in to theorem 2.3)
Theorem 2.3: If n is composite it has a prime divisor p satisfying 1  p  n .
Example: If d | ab does it follow that d | a or d |b ?
Theorem 2.4: If p is prime and p| ab then p| a or p|b . Prove.
Theorem 2.5: If p is prime and p| a1a 2  a k then p| a i for some i  1,2, k .
Theorem 2.6: (Unique Factorization Theorem) Any positive integer can be written
uniquely (up to order) as a product of primes.
Theorem 2.7: (Euclid) The Infinitude of Primes. Prove.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
3
Section 3: Linear Diophantine Equations
Recall: Mango Juice Problem. Consider these variations: 6x+10y=8, 6x+10y=2,
6x+10y=4, 6x+10y=3, 6x+10y=1. Which have solutions? Make a conjecture.
Theorem 3.1: The equation ax  by  c has integer solutions x , y if and only if
gcd (a , b)| c .
Example: Consider the Diophantine equations 2x+3y=1 and 6x+10y=8 (the mango problem).
Determine all solutions by connecting them to their linear graphs. Make a conjecture
(arrive at the following theorems)
Theorem 3.2: Suppose (a , b)  1 and x o , y o is a solution pair of ax  by  c . Then
(i) every solution is of the form x  x o  bt , y  yo  at , where t  Z
(ii) everything of this form is a solution
Verify (ii).
Theorem 3.3: Suppose (a , b)  d and x o , y o is a solution pair of ax  by  c . Then
(i) every solution is of the form x  xo  b d t , y  yo  a d t , where t  Z
(ii) everything of this form is a solution
Verify (ii) as homework problem.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
4
Section 4: Congruences
Example: What day of the week will it be 11 days from now? 95 days from now?
320772 days?
Definition: a congruent to b modulo m, written a  b mod m
Theorem 4.1: a  b mod m if and only if there exists an integer k such that
a  mk  b . Prove.
Theorem 4.2: Every integer is congruent mod m to exactly one of 0, 1, … , m  1 . Prove.
Definition: least residue
Alternate definition: a  b mod m if and only if a and b have the same remainder on
division by m .
Example: Prove d | a if and only if a  0 mod d
Theorem 4.4: For integers a, b, c, d
a) a  a mod m
b) If a  b mod m then b  a mod m
c) If a  b mod m and b  c mod m , then a  c mod m
d) If a  b mod m and c  d mod m , then a  c  b  d mod m
e) If a  b mod m and c  d mod m , then ac  bd mod m
Prove (e). Prove the remaining as an in-class exercise.
Examples: Compute 1) (71+59) mod 8;
2) (130 x 91) mod 11
3) (75+83 x 157– 5 x 53) mod 7
Example: T or F. If ac  bc mod m and (c, m)  1 then a  b mod m . Prove.
Examples: 1) solve 3x  9 mod 11
2) solve 3x  1 mod 11
Example: T or F. If ac  bc mod m does it follow that a  b mod m?
(No! One cannot cancel freely.)
Theorem 4.5: Every integer is congruent mod 9 to the sum of its digits. Prove.
Examples:
1) casting out nines
2) divisibility rules for 9 and 3
3) Lighting the Overhead on Fire problem from introduction
More divisibility rules.
Example: A correctly coded 10-digit ISBN a1a 2  a10 has the property that
10a1  9a 2  8a 3   2a 9  a10  0 mod 11.
1) this scheme detects all single-position errors
2) this scheme detects all transposition errors
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
5
Section 5: Linear Congruence Equations
Consider: General linear congruence equation ax  b mod m , where solutions are
congruence classes identified by the element in least residue (i.e. least residue
solutions). What can we expect when we solve them?
Example: Consider congruence equations 2 x  1 mod 6 , 2 x  1 mod 3 ,
2 x  2 mod 6 , 3x  7 mod 12 and 3x  6 mod 12 (recall Z12 table from
previous course) and make conjectures about when congruence equations have
no solutions, unique or multiple least residue solutions. If they have multiple
solutions, how many do they have?
Example: Compare congruence equation ax  b mod m to linear Diophantine equation
ax  my  b .
Theorem 5.1: The congruence equation ax  b mod m has solutions if and only if
(a , m)|b .
Theorem 5.2: If (a, m)  1 then ax  b mod m has exactly one least residue solution.
Theorem 5.3: If (a , m)  d and d |b then ax  b mod m has exactly d least residue
solutions.
Examples: 1) 6x  1 mod 13
2) 4 x  8 mod 10 (reduce to x  2 mod 5 to emphasize that sometimes the
equation and the modulus are divided by different numbers)
3) 5x  1 mod 12 (solve using multiplicative inverses from Z12 table)
4) 5x  11 mod 12 (solve using multiplicative inverses from Z12 table)
5) 7 x  8 mod 12 (solve using multiplicative inverses from Z12 table)
6) Can 8x  4 mod 12 be solved using multiplicative inverses? (Note when
coefficients do not have multiplicative inverses in mod m.)
Examples: For larger numbers some of the above approaches to solving congruence
equations are inefficient.
a) Consider 29 x  1 mod 83 . An efficient way to solve this is to find the
multiplicative inverse of 29 in mod 83: write as 29 x  83y  1 and compare
to Diophantine equations. Use Euclidean algorithm to find that x   20
and y  7 satisfy 29 x  83y  1 , which implies 29( 20)  1 mod 83
(a multiplicative inverse), so that x  63 mod 83 is the solution.
b) Now consider 29 x  17 mod 83 . We again use the fact that -20 is the
multiplicative inverse of 29 in mod 83 and multiply both sides of the
congruence equation 29 x  17 by (-20) and determine least residue
solution: x  75 mod 83.
c) 91x  25 mod 136
d) 132 x  25 mod 253 (no solution)
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
6
Section 6: Fermat’s and Wilson’s Theorems
Example: Compute the first six (6) powers of a in mod 7 for a = members of least
residue mod 7
Theorem 6.1: (Fermat’s Theorem) If p is prime and (a, p)  1 , then a p1  1 mod p .
Prove using the following lemma.
Lemma 1: If (a, m)  1 then the least residues of a , 2 a , 3 a , ,(m  1)a are
1, 2, 3, ,(m  1) . Explain with an example.
Example: (Fast exponentiation) Verify 7 30  1 mod 31
Theorem 6.2: (Wilson’s Theorem, if time) p is prime if and only if ( p  1)!   1 mod p
Give the idea of the proof after considering an example about multiplicative
inverses and Lemma 2.
Bonus Proof:
n
2 is irrational for n  2 (using Fermat’s Last Theorem)
a
for
b
some pair of integers a , b where b  0 . Take both sides of this equation to the nth power and
rearrange to obtain 2b n  a n . This equivalent to b n  b n  a n . Since n  2 this contradicts
Fermat’s Last Theorem. Thus n 2 is irrational.
By way of contradiction suppose that
n
2 is rational for any integer n  2 . Then
n
2
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
7
Section 7: Euler – function
Definition:  – function, written  (n)
Activity: Euler – function. Work in groups to arrive at formulas for calculating  (p) and
 (pq) for primes p and q.
Example: Application of  (n). Consider the equation a k  1 mod m (of which Fermat is
a special case). Build tables of integer powers of the least residue in various
moduli to form conjectures for criterion for a, k and m which satisfy
a k  1 mod m .
Theorem 7.1: (Euler’s Theorem) Given m  1 and (a, m)  1 then a  ( m)  1 mod m .
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
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