1. Modular Arithmetic
Lesson 4.6

2. This section, we are grouping numbers, based on their reminders.
These are said to be in the same congruence class.
18 ≡ 22 (mod 4)  bc their remainders are the same

3. How many congruence classes are there in modulo 7?
There are 7
congruence classes! R6 R0

4. Congruence Theorem
For all integers a and b and all positive integers m, a ≡ b (mod m) iff m is a factor of a – b.
ISBN and credit card check numbers are determined in some way by modular arithmetic.

5. Example 1
The final digit of a 12-digit Universal Product Code (UPC) is a check digit. Suppose the digits of a UPS are represented by
X1X2X3X4X5X6-X7X8X9X10X11Xc
To calculate the check digit Xc, first compute 3(X1+X3+X5+X7+X9+X11) + (X2+X4+X6+X8+X10) (mod 10). If this number is zero, then Xc=0, If not, subtract the number from 10.
Determine the check digit of – 09529
3( ) + ( )
= 3(18) + (22) =
76 ÷ 10 = 7 r. 6
So Xc=4

6. Example 2
Use modular arithmetic to explain why a date that falls on Friday this year will fall on Wednesday four years from now.
365 days in a year + 1 leap year!
1461 days ÷ 7 days in a week =
208 weeks r. 5
Friday – r. 0 Tuesday – r. 4
Sat. – r.1 Wednesday – r. 5
Sun. – r.2 Thursday – r. 6
Monday – r.3

7. Example 3 Find the smallest positive value of n for which
n – 32 ≡ 75 (mod 11)
n ≡ 107 (mod 11)
Congruence class of R8
n = 8

8. Homework
Pages 261 – 262
1-3, 5-11, 19