Modulo-m Systems

If m is an integer greater than 1, then a modulo-m system consists of the numbers 0,1,2,…,m−1. Counting and arithmetic operations are performed in a manner corresponding to movements on an m-hour clock. The number  is called the modulus of the system.

Let’s say we wanted to count to 53 using a modulus of 12. Let’s use a 12-hour clock to count to 53:

We can express the fact that 53 (a) occupies the same position as 5 (b) on a 12-hour clock more precisely. In that case, we say that a is congruent to b modulo-12, written a≡b (mod 12), provided that m evenly divides a-b. If you find it more convenient to divide m into b-a, this is also acceptable.

Q1.   Determine which statements are true.

To add, subtract, and multiply in a modulo-m system:

  1. Perform the operation as usual.
  2. Replace the result in (1) by one of the numbers  that is congruent to the result in part (1).

Q2.   Find:

  1. 7+4 (mod 8)
  2. 2−5 (mod 12)

Q3.   Solve 4+x≡2 (mod 5)