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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. 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

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  1. 4.1 Factors and Divisibility Remember to Silence Your Cell Phone and Put It In Your Bag!

  2. 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.

  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.

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