Session 5, Part B:
Divisibility Tests

In This Part: Developing Testing Rules | Divisibility Tests for 2, 5, and 10
Divisibility Tests for 3 and 9 | Divisibility Tests for 4 and 8 | Divisibility Test for 11

 Because 11 is one more than 10, the divisibility test for 11 is related to the test for 9. Remember that each power of 10 is one more than a multiple of 9. Some powers of 10 are also one more than a multiple of 11. For example, 1 is (0 • 11) + 1, and 100 is (9 • 11) + 1. Moreover, although 10 and 1,000 are not one more than a multiple of 11, they are one less than a multiple of 11. That is, 1,000 = (91 • 11) - 1, and 10 = (1 • 11) - 1. So what powers of 10 are one more than a multiple of 11? And what powers of 10 are one less than a multiple of 11? The base ten blocks below represent the number 1,111: To determine if 1,111 is divisible by 11, we express 1,111 as a sum: 1,000 + 100 + 10 + 1 = (1,001 - 1) + (99 + 1) + (11 - 1) + 1 This can be rewritten as: (1,001 + 99 + 11) + (-1 + 1 - 1 + 1), or 11 • (91 + 9 + 1) + 0. Thus, 1,111 is divisible by 11. Knowing this leads to the divisibility rule for 11. Here's the rule: Find the sum of the digits indicating odd powers of 10 (e.g., 101, 103, 105, etc.) and the sum of the digits indicating even powers of 10 (e.g., 100, 102, 104, etc.). If the difference between these two sums is divisible by 11, then the number is divisible by 11. In our example, we have (-1 + 1 - 1 + 1) which yields 0, and 0 is divisible by 11. There are divisibility tests for 7 as well, but the calculations involved take longer than dividing by 7! Note 3

 Problem B6 Use the divisibility test to determine if 11 divides 3,456.

Problem B7

Apply divisibility tests to find the missing digits so that

 a. 124,73_ is divisible by 9 b. 364,12_ is divisible by 33

 Problem B8 If the tests for both 2 and 6 work, can you assume that a number is divisible by 12? Explain.

 Problem B9 Devise a general rule for a divisibility test for 16.

 Problem B10 Devise divisibility tests for 12, 18, and 72.

Problem B11

 a. Devise a divisibility test for 3 in base four. b. Devise a divisibility test for 2 in base five.

Next > Part C: Factors

 Session 5: Index | Notes | Solutions | Video