7 - Number Theory

Summary

Divisibility

We say an integer b evenly divides another integer c if c/b is a whole number. Actually, nobody in mathematics ever says that b “evenly divides” c – people just say b “divides” c. Another way to say the same thing is to say that b is a divisor of c. The divisors of c are the numbers that (evenly) divide c. Finally, one can also say that b is divisible by c.

Examples:

1. • 3 divides 12.

2. • 3 is a divisor of 9.

3. • 40 is not a divisor of 20.

4. • 40 is divisible by 20.

5. • 4 divides 4.

6. • 5 is a divisor of −10.

7. • 12 divides 60.

8. • The positive divisors of 50 are 1, 2, 5, 10, 25, and 50.

Relative primality

Two numbers r and s are relatively prime if there is no integer bigger than 1 that is both a divisor of r and a divisor of s. We also say in this case that r is relatively prime to s. For example, 18 and 8 are not relatively prime because 2 is a divisor of both of them. On the other hand, 9 and 8 are relatively prime because the only divisors common to both of them are 1 and −1. We never count 1 and −1 as common divisors when determining relative primality.