Transcript Modular Arithmetic

MODULAR ARITHMETIC
Lesson 4.6

This section, we are grouping numbers, based on
their reminders.
18 ÷ 4 = 4 r. 2
22 ÷ 4 = 5 r. 2
78 ÷ 4 = 19 r. 2
These are said to be in the same congruence class.
18 ≡ 22 (mod 4)  bc their remainders are the same
HOW MANY CONGRUENCE CLASSES ARE
THERE IN MODULO 7?

77 ÷ 7 = 11 r. 0

78 ÷ 7 = 11 r. 1

79 ÷ 7 = 11 r. 2

80 ÷ 7 = 11 r. 3

81 ÷ 7 = 11 r. 4

82 ÷ 7 = 11 r. 5

83 ÷ 7 = 11 r. 6

84 ÷ 7 = 12 r. 0
R0
R1
R2
R3
R4
R5
R6
R0
77 ≡ 84 (mod 11)
There are 7
congruence classes!
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.
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 300219 – 09529
3(3 + 0 + 1 + 0 + 5 + 9) + (0 + 2 + 9 + 9 + 2)
= 3(18) + (22) = 76 ÷ 10 = 7 r. 6
So Xc=4
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

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
HOMEWORK
Pages
261 – 262
1-3, 5-11, 19