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    8.  A number has 8 and 9 as two of its factors.        How many times between 12:00 PM and 12:00
                         What other factors does that number have?           AM (midnight) will  the network experience
                                                                             a  slowdown due  to  all three  scripts  running
                               (a)  1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72  simultaneously?            (Critical Thinking)

                               (b)  1, 2, 3, 4, 6, 8, 9, 12, 18, 24                 (a)  1              (b)  2
                                                                                    (c)  3              (d)  4
                               (c)  1, 2, 4, 6, 8, 9, 18, 36, 72
                                                                         10.  A teacher is arranging seating for a class of
                               (d)  1, 3, 4, 6, 8, 12, 18, 36, 72            students in rows, with each row having 36,
                                                                             48, or 60 seats. The teacher wants to place
                      9.  A data center has three backup systems that        the students in such a way that after filling
                         run maintenance scripts.                            the seats in each row, there will be 9 extra
                            • System A runs every 42 minutes.                students left over. What is the minimum
                                                                             number of students the teacher needs to
                            • System B runs every 60 minutes.                arrange so that, when divided by 36, 48, and
                            • System C runs every 70 minutes.                60,  there will  always  be  exactly  9  students
                                                                             remaining?                 (Critical Thinking)
                         All three scripts were triggered at 12:00 PM.
                         A  technician  notices  that  the  network  slows          (a)  711            (b)  720
                         down significantly  only when at  least  two
                         scripts run at the exact same time.                        (c)  729            (d)  738

                                       We are given that three numbers—A, B, and C—satisfy the following conditions:

                                         1.  The highest common factor (HCF) of any two of these numbers is the same
                                            prime number (p). Specifically:

                                                 • The HCF of A and B is (p).
                                                 • The HCF of B and C is (p).
                                                 • The HCF of A and C is (p).
                                         2.  The least common multiple (LCM) of all three numbers (A, B, and C) is

                                            p × p × q × r .
                                         3.  Each of A, B, and C is a composite number, and no two numbers are equal.

                                       What is the smallest possible value for the sum A + B + C in terms of p, q and r.

                                             (a)  r(p + p + pr)                     (b)  q(p + r + pr)

                                             (c)  p(q + r + qr)                     (d)  p(pr + q)

                                                                                         Finding Common Ground        45

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