Introduction to Factors, Divisors, and Integers

Look at a case where it's not a factor: a divisor must divide the dividend exactly so that the quotient is an integer.
Example: five divided by 10
- \frac{5}{10} = \frac{1}{2} = 0.5
- Since the quotient is not an integer, 10 is not a divisor of 5.
- Both 5 and 10 are integers, but the division does not yield a whole number.
Another quick check: 5 divided by 2
- \frac{5}{2} = 2.5
- Not an integer; therefore 2 is not a divisor of 5.
What is a divisor? A divisor (or factor) of a number N is an integer d such that N ÷ d is an integer, i.e., d | N. For example, 3 | 15 because 15 ÷ 3 = 5.
Important notation:
- If a | N, then a is a divisor (factor) of N.
- Example: 5 | 5 and 1 | 5 (and negative divisors as well: -1 | 5, -5 | 5), but we often list positive divisors for simplicity.

Recap: the factors of 5 are the integers that divide 5 without leaving a remainder.
Positive divisors: {1, 5}
Including negatives (also divisors): { -5, -1, 1, 5 }
Factor pairs for 5 can be seen as: 5 = 1 \cdot 5 = (-1) \cdot (-5), with corresponding sign variations.
Full statement in divisibility terms: If a \mid 5, then there exists an integer k with 5 = a \cdot k. For the divisors of 5, the valid pairs are:
- 5 = 1 \cdot 5
- 5 = (-1) \cdot (-5)
- and similarly 5 = 5 \cdot 1, 5 = (-5) \cdot (-1).
Summary: The divisors of 5 are { -5, -1, 1, 5 }; the positive divisors are {1, 5}.
Quick practice examples (using the concept of divisibility):
- Does 3 divide 15? yes, since 15 ÷ 3 = 5\, and 3 \mid 15;
- Does 6 divide 15? no, since 15 ÷ 6 = \tfrac{5}{2} and not an integer; 6 \nmid 15.
- Divisors of 6: { -6, -3, -2, -1, 1, 2, 3, 6 }; positive divisors: {1, 2, 3, 6}.
Additional notes:
- A divisor of N is a number that multiplies with some integer to give N: N = d \cdot q for some integer q.
- The quotient when dividing N by a divisor d is q = N/d, which must be an integer for d to be a divisor.