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                     \ 09-Nov-2025  Bharat Arora   Proof-8                                    Reader’s Sign _______________________ Date __________

                       4.  Check if the first number is a factor of the second number.
                           a.  8, 176               b.  12, 140              c.  10, 160               d.  15, 220

                      5.  Find the numbers between 1 and 30 which have only:

                           a.  two factors ____  b.  four factors ____  c.  three factors ____
                      6.  Say whether the following statements are true or false.

                           a.  5 is a factor of 70.                       b.  8 is a factor of 92.
                           c.  1 is a factor of 121.                      d.  0 is a factor of 28.

                           e.  1 has two factors.                         f.   32 is a factor of itself.

                    COMMON FACtORS

                    The numbers which are common among the factors of                      1 × 12  = 12      1 × 30  = 30

                    two or more numbers are called common factors.                         2 × 6  = 12       2 × 15  = 30

                    Let us consider any two numbers, say 12 and 30.                        3 × 4  = 12       3 × 10  = 30
                    Factors of 12 are 1, 2, 3, 4, 6 and 12.                                Factors           5 × 6  = 30

                    Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.                                           Factors

                    We can see that the factors 1, 2, 3 and 6 are present in both lists.
                    So, the common factors of 12 and 30 are 1, 2, 3 and 6.

                    Finding Common Factors

                    To find the common factors of two or more numbers, the following steps are taken:

                        1.  Write all the factors of the first number.

                      2.  Write all the factors of the second number.

                      3.  Highlight or circle the factors which are common in both the lists.

                    example 1. Find the common factors of 8 and 10.

                    Solution. 8 can be written as:                    10 can be written as:            Common factors

                                    8 = 1 × 8                                10 = 1 × 10
                                                                                                               1    5
                                    8 = 2 × 4                                10 = 2 × 5
                                        Factors                                  Factors                 4     2
                                                                                                         8          10
                    The factors of 8 are 1, 2, 4 and 8.
                                                                                                       Factors  Factors
                    The factors of 10 are 1, 2, 5 and 10                                                of 8      of 10

                    So, the common factors of 8 and 10 are 1 and 2.

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                                                                                           Multiples and Factors

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