Download Section 4 Notes - University of Nebraska–Lincoln

SECTION 4: Congruences
Comments to the Instructor
“Hassengabel” is a German game (literally meaning “hates to divide”) to determine who among
a group of people should get the one remaining piece of pie (or other desirable item). Before
starting, someone is designated as the counter. All players stand in a circle. Then, the players
say “eins, zwei, drei” (with each number you hit the palm of one hand with the fist of the other,
similar to rock, paper, scissors) and on “drei” they display between 0 and 5 fingers. The counter
adds up all of the fingers, and starting with the person on his or her left, counts clockwise until
the sum of the fingers is reached. The person where the count ends is the winner of the piece of
pie. This is a good example of how modular arithmetic can be used (especially if you decide to
allow between 0 and 10 fingers) to predict where the count will end without having to count
one-by-one until the sum is reached. Consider playing the game at the conclusion of the lesson.
Nice connections to middle level mathematics curriculum can be made when discussing
divisibility rules, prime factorization, greatest common divisors, and other number bases.
A. INSTRUCTOR MATERIALS
 Video: “The Proof” (by Nova, 60 minutes)
 ISBN worksheet
B. PARTICIPANT MATERIALS
 ISBN worksheet
 calculators
 Handout: Section 4 outline
C. SESSION NOTES
Section 4: Congruences
Example: What day of the week will it be 11 days from now? 95 days from now?
320772 days? (Draw boxes to represent the days of the week and write a variety of
numbers that fit into each box. Then connect the boxes to congruence classes noting that
although each box has a name – the least residue element – there are many numbers in
each box.)
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
Theorem 4.3 (Alternate definition): a  b mod m if and only if a and b have the same
remainder upon division by m .
Activity: Have students number off from 1, 2,… 31. Write 0, 1,…, 6 at different places in
the room and ask students to go to the “box” which gives their remainder after
dividing by 7. Repeat for mod 2 using a zero and one box.
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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*Activity: Say “n is odd” in three ways. 1) n = 2k+1 for some integer k, 2) n  1mod 2,
3) n has a remainder of 1 when divided by 2.
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+69) mod 8 2) (130x91) mod 11
3) 75+83 x 157-5 x 53) mod 7
4) solve x  8  7 mod 19
Example: T or F. If ac  bc mod m does it follow that a  b mod m?
(Cannot cancel freely!)
(*Theorem: If ac  bc mod m and (c,m)=1 then a  b mod m.)
(*Theorem: If ac  bc mod m and (c,m)=d then a  b mod m/d.)
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) the Penny Problem from introduction
More divisibility rules (note rules for 4, 8, 11 are in homework)
Example: (See ISBN worksheet) A correctly coded 10-digit ISBN a1a 2  a10 has the
property that 10a1  9a2  8a3   2a9  a10  0 mod 11.
1) this scheme detects all single-position errors
2) this scheme detects all transposition errors
* indicates material that may be omitted due to time constraints
Work Problem Session 4 problems in class.
D. ASSIGNMENT
 Work Evening Homework problems for Section 4
 Reading: The Mathematical Universe, Chapter P
 Work Reading Assignment Problems for Chapter P
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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