5.1 Number Theory 223 Perfect Numbers A number whose proper factors (factors other than the number itself) add up to the number is called a perfect number. For example, 6 is a perfect number because its proper factors are 1, 2, and 3, and 1 2 3 6. + + = Determine which, if any, of the following numbers are perfect numbers. 79. 28 Yes 80. 36 No 81. 72 No 82. 496 Yes Challenge Problems/Group Activities In Exercises 83–84, determine (a) the greatest common divisor (GCD) and (b) the least common multiple (LCM). 83. 24, 48, and 128 a) 8 b) 384 84. 18, 78, and 198 a) 6 b) 2574 85. Number of Positive Factors The following procedure can be used to determine the number of positive factors (or divisors) of a composite number. Write the number in prime factorization form. Examine the exponents on the prime numbers in the prime factorization. Add 1 to each exponent and then determine the product of these numbers. This product gives the number of positive factors of the composite number. For example, the prime factorization of 75 is 3 5 .2 ⋅ The exponents on the prime factors are 1 and 2. Therefore, 75 has (1 1) (2 1) 2 3 + ⋅ + = ⋅ or 6 positive factors (the factors are 1, 3, 5, 15, 25, and 75). a) Use this procedure to determine the number of positive factors of 60. 12 b) To check your answer, list all the positive factors of 60. You should obtain the same number of factors determined in part (a). 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 86. Recall that if a number is divisible by both 2 and 3, then the number is divisible by 6. If a number is divisible by both 2 and 4, is the number necessarily divisible by 8? Explain your answer. * 87. The product of any three consecutive natural numbers is divisible by 6. Explain why. * 88. A number in which each digit except 0 appears exactly three times is divisible by 3. For example, 888,444,555 and 714,714,714 are both divisible by 3. Explain why this outcome must be true. The sum of the digits of such numbers is divisible by 3; therefore, the number itself is divisible by 3. 89. Use the fact that if a b| and a c, | then a b c ( ) | + to determine whether 54,036 is divisible by 18. (Hint: Write 54,036 as 54,000 36. + ) Yes 90. Mersenne Primes Show that 2 1 n − is a (Mersenne) prime for n 2, 3, 5, = and 7 but composite for n 11. = 3, 7, 31, and 127 are all prime, but 2047 is divisible by 23 and 89. 91. Another Conjecture Goldbach also conjectured in his letter to Euler that every integer greater than 5 is the sum of three prime numbers. For example, 6 2 2 2 = + + and 7 2 2 3. = + + Show that this conjecture is true for integers 8 through 20. * 92. Divisibility by Seven The following describes a procedure to determine whether a number is divisible by 7. We will demonstrate the procedure with the number 203. i) Remove the units digit from the number, double the units digit, and subtract it from the remaining number. For 203, we will remove the 3, double it to get 6, and then subtract 20 6 − to get 14. ii) If this new number is divisible by 7, then so is the original number. In our case, since 14 is divisible by 7, then 203 is also divisible by 7. iii) If you are not sure if the new number obtained in Step ii is divisible by 7, you can repeat the process described in the first step. Use the procedure described above to determine whether the following numbers are divisible by 7. a) 329 Yes b) 553 Yes c) 583 No d) 4823 Yes 93. Prime Numbers Consider the first eight prime numbers greater than 3. The numbers are 5, 7, 11, 13, 17, 19, 23, and 29. a) Determine which of these prime numbers differs by 1 from a multiple of the number 6. 5, 7, 11, 13, 17, 19, 23, and 29 b) Use inductive reasoning and the results obtained in part (a) to make a conjecture regarding prime numbers. Every prime number greater than 3 differs by 1 from a multiple of 6. c) Select a few more prime numbers and determine whether your conjecture appears to be correct. This conjecture should appear to be correct. Research Activities 94. GIMPS (See Mathematics Today on page 220.) Write a report on the GIMPS project. Describe the history and development of the project. Include a current update of the project’s findings. 95. Deficient and Abundant Numbers Explain what deficient numbers and abundant numbers are. Give an example of each type of number. *See Instructor Answer Appendix
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