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§5.1 Divisibility Tests 02/19/17 Today We’ll Discuss What does it mean for a number to be divisible by another? What are some common divisibility tests? Definition of Divisibility Definition: Let a and b be whole numbers such that a ≠ 0. We say a divides b if and only if there is a whole number n such that b = an. We write this as a∣b. Definition of Divisibility WAIT!! What does if and only if mean? If a divides b then there exists a whole number n such that b = an. If there exists a whole number n such that b = an then a divides b. Definition of Divisibility For a|b we can say: • a is a factor of b • b is a multiple of a • b is divisible by a • b is evenly divisible by a (“Does not divide” is written as a∤b.) Example a) Use the definition of divisibility to state why 3 divides 12. b) Use the definition of divisibility to state why 8 divides 1256. Example a) Give an example of a multiple of 12. b) Give an example of a factor of 60. c) True or False: 20∣4 d) Give an example of a number that is divisible by 2 but not 4. Divisibility Tests for 2, 5, 10 Test for 2: A whole number is divisible by 2 if and only if its ones digit is an even number. (0,2,4,6,8) Test for 5: A whole number is divisible by 5 if and only if its ones digit is 0 or 5. Test for 10: A whole number is divisible by 10 if and only if its ones digit is 0. Properties of Divisibility a, b, m, and n are whole numbers (n ≠ 0) Sum Property If n∣a and n∣b, then n∣(a + b) Difference Property If n∣a and n∣b, then n∣(a − b) Properties of Divisibility a, b, m, and n are whole numbers (n ≠ 0) Sum Property If n∣a and n∣b, then n∣(a + b) Difference Property If n∣a and n∣b, then n∣(a − b) Product Property If n∣a, then n∣am Example Use the divisibility properties to complete each statement. a) 2∣4 and 2∣12, so … b) 5∣15 and 5∣35, so… c) 4∣32, so 4∣(5•__) Developing Divisibility by 4 Do you know the divisibility test for 4? Let’s take a look at how to discover it. Divisibility Tests for 4, 8 Test for 4: A whole number is divisible by 4 if and only if the last two digits of the whole number is divisible by 4. Test for 8: A whole number is divisible by 8 if and only if the last three digits of the whole number is divisible by 8. Example Determine whether or not the following are divisible by 4 or 8: a) 1450 b) 2728 c) 376 d) 1800 Developing Divisibility by 3, 9 Do you know the divisibility tests for 3 or for 9? Let’s take a look at how to discover them. Divisibility Tests for 3, 6, 9 Test for 3: A whole number is divisible by 3 if and only if the sum of all its digits is divisible by 3. Test for 6: Passes tests for divisibility of both 2 and 3. Test for 9: A whole number is divisible by 9 if and only if the sum of all its digits is divisible by 9. Example Determine whether or not the following are divisible by 3, 6, or 9. a) 1375 b) 2094 c) 3744 d) 1750 Discussion If n∣(a + b), is it always true that n∣a and n∣b? Homework #10 §5.1 Pages 166-169 (Problems #7-21 You do not have to explain but you should be able to on a quiz or exam!) #7,12,13,16,18,20,21,24,36,39