Integer Problems                                          Up

         The set of integers is made up of the set of positive and negative whole numbers
        including 0.         

           Any integer with a factor 2 is an even integer. The formula for any even
           integer is 2n, where n is any integer.
           The formula for any odd integer is 2n + 1 or 2n - 1.
           N,  N + 1,  N + 2,  N + 3,... is the set of consecutive numbers.
           2N, 2N + 2, 2N + 4, 2N + 6,... is the set of consecutive even integers.
           2N + 1, 2N + 3, 2N + 5,... is the set of consecutive odd integers.
          If X is the integer divisible by another integer Y, then Y is said to be a factor
          of X, or X is said to be multiple of Y.
          Example:   Since 6 is divisible by 2 and 3, they are the factor of 6. 
          A number that has no other factors but itself and 1 is called a prime number.
          Example:   7, 11, 19 are examples of prime numbers. Numbers which are not prime
          called composite. Thus, 15 is a composite number. 15 = 135.
          There are the certain rules we have to know working with the Integer Problems.
                   Addition                                                    Multiplication
          even + even = even                                       even even = even
          odd + odd = even                                          even odd = even
          even + odd = odd                                          odd odd = odd
         If you don't remember these rules, you can choose your own numbers to check
         the result.

             Example:  If p and r are different prime numbers greater than 9, which of
           the following  must be true ?
            I.      is less than 1                  II.     p + r  is not prime
           III.  p r  has three different positive integer factors greater than 1
               Answers: A) None;  B) III only;  C) I and II;  D) I and III;  E) II and III
                    Solution
        1. Let  p = 13  and  r = 11.  Then  , so choice I is incorrect.
        2. Since any prime number greater than 9 is odd ( the even prime number is
          2 only ), then sum of two odd numbers is even, and not a prime, so choice II is
           correct.
        3. Check the choice III:  p r has three different integer factors greater than 1
            ( p, r and the product of these two p r ). Example: 13,11 and 13 11 = 143
            So choice III is correct.
                      The answer is E.

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