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Chapter 4 Number Theory Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 4-1 Divisibility Divisibility Divisibility Rules Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility A whole number is even if it has a remainder of 0 when divided by 2; it is odd otherwise. We say that “3 divides 18”, written 3 | 18, because the remainder is 0 when 18 is divided by 3. Likewise, “b divides a” can be written b | a. We say that “3 does not divide 25”, written , because the remainder is not 0 when 25 is divided by 3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility In general, if a is a whole number and b is a nonzero whole number, we say that a is divisible by b, or b divides a if and only if the remainder is 0 when a is divided by b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition If a and b are any whole number, then b divides a, written b | a, if, and only if, there is a unique whole number q such that a = bq. If b | a, then b is a factor or a divisor of a, and a is a multiple of b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-1 Classify each of the following as true or false. a. −3 | 12 True c. 0 is even. True b. 0 | 2 False d. True e. For all whole numbers a, 1 | a. f. True For all non-zero whole numbers a, a2 | a5. True g. 3 | 6n for all whole numbers n. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. True Example 4-1 h. 3 | (5 7 9 11 1) g. 0 | 0 (continued) True False Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Properties of Division For any whole numbers a and d, if d | a, and n is any whole number, then d | na. In other words, if d is a factor of a, then d is a factor of any multiple of a. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Properties of Division For any whole numbers a, b, and d, d ≠ 0, a. If d | a, and d | b, then d | (a + b). b. If d | a, and , then c. If d | a, and d | b, then d | (a − b). d. If d | a, and , then Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-2 Classify each of the following as true or false, where x, y, and z are whole numbers. a. If 3 | x and 3 | y, then 3 | xy. True b. If 3 | (x + y), then 3 | x and 3 | y. False c. False Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility Rules Sometimes it is useful to know if one number is divisible by another just by looking at it. For example, to check the divisibility of 1734 by 17, we note that 1734 = 1700 + 34. We know that 17 | 1700 because 17 | 17 and 17 divides any multiple of 17. Furthermore, 17 | 34; therefore, we conclude that 17 | 1734. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility Rules Another method to check for divisibility is to use the integer division button INT ÷ on a calculator. Press the following sequence of buttons: 1 7 3 4 INT ÷ to obtain the display 102 Q 1 7 0. R Copyright © 2013, 2010, and 2007, Pearson Education, Inc. = Divisibility Tests A whole number is divisible by 2 if and only if its units digit is divisible by 2. A whole number is divisible by 5 if and only if its units digit is divisible by 5, that is if and only if the units digit is 0 or 5. A whole number is divisible by 10 if and only if its units digit is divisible by 10, that is if and only if the units digit is 0. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility Tests A whole number is divisible by 4 if and only if the last two digits of the number represent a number divisible by 4. A whole number is divisible by 8 if and only if the last three digits of the whole number represent a number divisible by 8. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-4a Determine whether 97,128 is divisible by 2, 4, and 8. 2 | 97,128 because 2 | 8. 4 | 97,128 because 4 | 28. 8 | 97,128 because 8 | 128. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-4b Determine whether 83,026 is divisible by 2, 4, and 8. 2 | 83,026 because 2 | 6. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility Tests Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility Tests A whole number is divisible by 3 if and only if the sum of its digits is divisible by 3. A whole number is divisible by 9 if and only if the sum of the digits of the whole number is divisible by 9. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-6a Determine whether 1002 is divisible by 3 and 9. Because 1 + 0 + 0 + 2 = 3 and 3 | 3, 3 | 1002. Because Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-6b Determine whether 14,238 is divisible by 3 and 9. Because 1 + 4 + 2 + 3 + 8 = 18 and 3 | 18, 3 | 14,238. Because 9 | 18, 9 | 14,238. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-7 The store manager has an invoice for 72 calculators. The first and last digits on the receipt are illegible. The manager can read $■67.9■ What are the missing digits, and what is the cost of each calculator? Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-7 (continued) Let the missing digits be represented by x and y, so that the number is x67.9y dollars, or x679y cents. Because 72 calculators were sold, the amount must be divisible by 72. Because 72 = 8 · 9, the amount is divisible by both 8 and 9. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-7 (continued) For the number on the invoice to be divisible by 8, the three-digit number 79y must be divisible by 8. Only 792 is divisible by 8, so y = 2, and the last digit on the invoice is 2. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-7 (continued) Because the number on the invoice must be divisible by 9, we know that 9 must divide x + 6 + 7 + 9 + 2, or x + 24. Since 3 is the only single digit that will make x + 24 divisible by 9, x = 3. The number on the invoice must be $367.92. The calculators cost $367.92 ÷ 72 = $5.11, each. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Divisibility Tests A whole number is divisible by 11 if and only if the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11. An whole number is divisible by 6 if and only if the whole number is divisible by both 2 and 3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 4-8 The number 57,729,364,583 has too many digits for most calculator displays. Determine whether it is divisible by each of the following: a. 2 No b. 3 No c. 5 No d. 6 No e. 8 No f. No g. 10 No 9 h. 11 Yes Copyright © 2013, 2010, and 2007, Pearson Education, Inc.