SECTION 1: Integers and Divisibility
Comments to the Instructor
Participants of previous courses have found the formal proofs on day one to be very difficult.
They were generally uncomfortable writing them and even those who seemed to have grasped the ideas were far from confident in their work (some were even angry that we were expecting this of them). However, when we returned to writing formal proofs on day 3, the majority of the participants expressed a significantly improved understanding (and attitude) regarding proofwriting. A strategy for bringing about this transformation: 1) Develop the first proof written in class (Theorem 1.1) with extreme care . Participants will follow this model much like a recipe until their comfort level increases (which means that some will follow the “recipe” for the remainder of the course!). 2) Have participants work in groups up at the board (simultaneously) to prove Theorems 1.2 and 1.3 (the in-class exercises). Be very particular and deliberate about language use and the use of symbols. Make note of how the omission of a single symbol/phrase
(like “there exists” or “for some”) can dramatically change the meaning of the proof.
It is also very important to include an activity involving even and odd integers. The need to apply the definitions comes up repeatedly throughout the course. They also provide an understandable forum in which to practice proofs.
The algebra in the Extended Euclidean Algorithm used to determine integers x and y for a given a and b (and their greatest common divisor d ) so that ax + by = d proved to be difficult for some of the teachers. However, successfully determining the integers which satisfy the equation is a vital part of the course (for solving congruence equations in section 5 and making sense of RSA cryptography) so this section should not be “hurried”.
A.
INSTRUCTOR MATERIALS
 set of 20 pennies for the penny problem
 video: “N is a Number” (by Nova, 60 minutes)

Remember to order cakes in preparation for the last class session! (See RSA Activity for details)
B.
PARTICIPANT MATERIALS
 course notebook
 calculators

Section 1 outline (see Course Notebook & Handouts folder)
C.
SECTION NOTES
Introduction
The Penny Problem
Fountain of Knowledge problem
Review of proof and logic notation
Review of basic properties for integers and operations: closure of +, – ,

; distributive, associative, commutative properties
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 c a
1
 c b
2
Theorem 1.4e: Given d a
1
, c i

Z, i d a
2
, … and d a n
,

1 2  n .
for c c
1 2

Z . Prove as in-class exercise.
then d c a
1 1
 c a
2 2 c a n n
) for
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 ( , )
Examples (using notation): Find (80, 12), (121,136), (1234, 34560)
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
  b that a
 bq
 r
Theorem 1.6: If a
 bq
 r
.
then ( , )

( , ) .
such
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 (optional, see pg 6, Dudley and
miscellaneous section of the notebook)
Key: a = 18cm, b
1
= 12cm, b
2
= 10cm, b
3
= 5cm
Theorem 1.7: If ( , )
 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
2) 80x + 12y = 4
4) 1234x + 34560y = 2
5) 231x + 20y = 1
3) 121x + 136y = 1
Theorem 1.8: If d ab and ( , )

1 then d b
Theorem 1.9: If a m
. Prove.
and b m and ( , )

1 then ab m . *Prove.
* indicates material that may be omitted due to time constraints
Work Problem Session 1 problems in class (in groups). Discuss.
Even and Odd Numbers (activity)
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
2

Arrive at definitions of even and odd integers. Prove together that (1) the sum of an even integer and an odd integer is odd; and (2) the product of an even integer and an odd integer is even.
Have students work in groups to prove a few other results about sums and products of even and odd integers.
D.
ASSIGNMENT

Evening Homework problems for Section 1

Reading: The Mathematical Universe , Chapter F

Reading Assignment Problems for Chapter F
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
3