Download 4.1 Factors and Divisibility

4.1 Factors and Divisibility
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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.