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:

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

**Q2.** Find:

- 7+4 (mod 8)
- 2−5 (mod 12)

**Q3.** Solve 4+x≡2 (mod 5)

**Solutions:**