Chipola College
MGF 1107
10.3 Modular Arithmetic__________________________________________________
A modulo m system consists of m elements, 0 through m-1, and a binary operation.
For example, the modulo 8 system consists of the elements {0,1,2,3,4,5,6,7}.
In any modulo system we can develop a set of modulo classes by placing all numbers
with the same remainder when divided by m in the appropriate modulo class. Our
modulo classes come from the elements in our modulo m system. For example, in our
modulo 8 system, we have the classes 0,1,2,3,4,5,6,7. The sets are composed in the
following manner:
0 class = {…,-16,-8,0,8,16,24,…}
1 class = {…,-15,-7,1,9,17,25,…}
2 class = {…,-14,-6,2,10,18,26,…}
3 class = {…,-13,-5,3,11,19,27,…}
4 class = {…,-12,-4,4,12,20,28,…}
5 class = {…,-11,-3,5,13,21,29,…}
6 class = {…,-10,-2,6,14,22,30,…}
7 class = {…,-9,-1,7,15,23,31,…}
An integer a is congruent to an integer b modulo m, written a≡b (mod m), if a has
remainder b when divided by m.
For instance, 23≡7 (mod 8) since 23 divided by 8 gives remainder 7.
Determine the modulo class in each of the following:
1. 46≡? (mod 3)
2. 112≡? (mod 13)
3. -13≡? (mod 37)
a. 18≡? (mod 7)
b. 42≡? (mod 7)
c. 12≡? (mod 12)
e. 144≡? (mod 32)
f. -17≡? (mod 8)
g. 1001≡? (mod 35)
YOU TRY:
Chipola College
MGF 1107
10.3 Modular Arithmetic__________________________________________________
We can also add, subtract, and multiply in modulo systems. In order to do this, we
compute the solution using the given operation, and then we find which modulo class the
solution belongs to.
Compute the following:
4. 9-1 in mod 6
5. 14+18 in mod 15
6. 11*12 in mod 15
a. 675-236 in mod 20
b. 17-20 in mod 9
c. 55+19 in mod 18
e. 1050+67 in mod 25
f. 6*7 in mod 20
g. -2*-5 in mod 7
YOU TRY:
For the following exercises, assume that Sunday is represented by day 0, Monday is
represented by day 1, and so on. If today is Thursday (day 4), determine the day of the
week it will be at the end of each period. Assume no leap years.
1. 161 days
2. 2 years
YOU TRY:
a. 463 days
b. 3 years, 27 days