1. 4.1 Factors and Divisibility
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2. Definition of Factor and Multiple
Factor - Any of the numbers or symbols in mathematics that when multiplied together form a product.
Multiple - The product obtained when multiplying a number by a whole number.
If a, b  W and a  b = c, then a is a factor of c, b is a factor of c, and c is a multiple of both a and b.

3. Factors Be able to find all factors of a number
Factor Test Theorem
To find all the factors of a number n, test only those natural numbers that are no greater than the square root of the number.
A natural number that has an odd number of factors is called a square number or square.

4. Definition - Divisibility
For a, b,  W, a  0, a divides b, written a | b, iff there is a whole number x so that a  x = b.
a is a divisor of b
b is divisible by a
a | b means that a does not divide b

5. Divisibility Tests
You are responsible for knowing the divisibility tests for 2, 3, 4, 5, 6, 9, and 10

6. Divisibility Tests
n is divisible by 2 iff its units digit is 0, 2, 4, 6, or 8
n is divisible by 3 iff the sum of its digits is divisible by 3
n is divisible by 4 iff the number represented by its last two digits is divisible by 4
n is divisible by 5 iff its units digit is 0 or 5

7. Divisibility Tests (cont.)
n is divisible by 6 iff it is divisible by both 2 and 3
n is divisible by 9 iff the sum of its digits is divisible by 9
n is divisible by 10 iff the units digit is 0

8. Definition of Even and Odd Numbers
A whole number is even iff it is divisible by 2.
A whole number is odd iff it is not divisible by 2.

9. Divisibility Theorems
For a, b, c, n  N
If a | b and a | c, then a | (b + c).
If a | b and a | c, then a | (b – c).
If a | c, b | c, and a and b have no common factors except 1, then a  b | c.
If a | b, then a | n  b.
If a | b, then (b  a) | b.

10. Divisibility Theorems (cont.)
If a | (b + c) and a | b, then a | c.
If a | (b – c) and a | b, then a | c.