Math 130

Fall 03

Primes, Factors, and Multiples

 

Practice exercises (answers attached):

 

1) Factor into primes:

            a) 50    b) 34    c) 800  d) 72    e) 32    f) 67     g) 555    h) 79  i) 1440

 

2) List all the factors:

            a) 50   b) 34   c) 25   d) 72    e) 137   f) 51

 

3) The Greatest Common Factor (GCF) of two numbers is the largest number that is a factor of both.  For example, the GCF of 15 and 20 is 5, since 15 has 1, 3, 5, and 15 as factors and 20 has 1, 2, 4, 5, 10, and 20 as factors, and 5 is the largest number that is on both lists.  Find the GCF of each pair of numbers:

 

            a) 10 and 16      b) 28 and 77      c) 36 and 48       d) 200 and 300

 

4) The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both the numbers.  For example, let's look at the numbers 15 and 20 again.  The first few multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120, 135.  The first few multiples of 20 are 20, 40, 60, 80, 100, 120, 140.  The smallest number on both lists is 60, and the LCM of 15 and 20 is 60.  Find the LCM of each pair of numbers:

 

            a) 4 and 6    b) 10 and 16    c) 20 and 30     d) 200 and 300    

 

5) Factor each of the following into primes and look for patterns in the prime factorizations: 10, 100, 1000, 10000, 1000000.  What is the prime factorization of a googol (1 followed by 100 zeroes)?

 

Problems:

 

1)  Find the smallest prime number greater than 1000.  Prove that your number is both prime and the smallest such prime.  You can use a spreadsheet to save time (make up an equation that will easily “check” many possible factors).

 

2)  Which natural number less than 100 has the most factors?  How could you convince a skeptic without factoring all the numbers?

 

3) Two natural numbers are called relatively prime if they have no common factor greater than 1.  Find two relatively prime natural numbers that are each greater than 1000.

 

4)  Find a relationship between the least common multiple and greatest common factor of two natural numbers.

 

5) Fill in table below and look for patterns.  This exercise will be useful when we study divisibility rules.

 

Num                 Divisible by 9         Divisible by 11         Num     Div by 9           Div by 11                                 (Yes or No)          (yes or no)                                 (Y/N)              (Y/N)

 

              9              yes                          no                      11      no                   yes

            99                                                                    101

          999                                                                   1001

        9999                                                                 10001

      99999                                                               100001

    999999                                                             1000001

  9999999                                                           10000001

99999999                                                        100000001

 

 

 

 

 

 

 

Answers to exercises:

            1a) 2 x 52  b) 2 x 17  c) 25 x 52   d) 23 x 32  e) 25   f) 67  g) 3 x 5 x 37  h) 79 

               i)25 x 32 x 5

 

            2 a) 1,2,5,10,25,50    b) 1,2,17,34   c) 1,5,25   d) 1,2,3,4,6,8,9,12,18,24,36,72

                e) 1,137   f) 1,3,17,51

 

            3 a) 2   b) 7   c)  12   d) 100

 

            4 a) 12   b) 80  c) 60   d) 600

 

            5) 10 = 2 x 5

                100 = 22 x 52

                 1000 = 23 x 53

                10000 = 24 x 54

                  100000 = 25 x 55, and a googol = 2100 x 5100

 

 

 

   Copyright 2005, Debra K. Borkovitz.  You may copy or edit this material for non-profit, educational use only.

 

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