4.1 Factors and Divisibility Remember to Silence Your Cell Phone and Put It In Your Bag!

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.

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.

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

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

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

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

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.

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.

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