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)
YOU TRY:
a. 18≡? (mod 7) b. 42≡? (mod 7) c. 12≡? (mod 12)
e. 144≡? (mod 32) f. -17≡? (mod 8) g. 1001≡? (mod 35)
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
YOU TRY:
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
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