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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.

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4 1 factors and divisibility

4.1 Factors and Divisibility

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

definition of factor and multiple
Definition ofFactor 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
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
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
Divisibility Tests

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

divisibility tests1
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
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
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
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
Divisibility Theorems (cont.)
  • If a | (b + c) and a | b, then a | c.
  • If a | (b – c) and a | b, then a | c.