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

                    Here, we can see that 18 buttons can be arranged in a rectangular arrangement in
                    3 ways: 1 × 18, 2 × 9 and 3 × 6.

                    Numbers 1, 2, 3, 6, 9 and 18 are termed as factors of 18.

                    Thus, we can say that when we multiply two numbers to get a product, the numbers

                    being multiplied are called factors of the product.

                    example 1. Find the factors of 12.

                    Solution.  We can see that 12 can be found by multiplying 1 and                          1  ×  12   =  12

                    12; 2 and 6; and 3 and 4. Thus, the numbers 1, 2, 3, 4, 6 and 12 are                     2  ×  6   =  12
                    factors of 12.                                                                           3  ×  4   =  12

                    properties of Factors

                      1.  Every number has 1 as its smallest factor.
                          For example, 1 × 3 = 3, 1 × 6 = 6, 1 × 9 = 9, 1 × 12 = 12, etc.

                      2.  The biggest factor of a number is the number itself.

                          For example, 6 × 1 = 6, 8 × 1 = 8, 16 × 1 = 16, 18 × 1 = 18, etc.

                      3.  Any factor of a number is either equal to or less than the number.
                          For example, the factors of 16 are 1, 2, 4, 8 and 16.

                      4.  There are a finite number of factors of a number.

                          For example, the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.
                          So, the factors are limited.

                      5.   Every number  except  1 has at  least  two  factors—1 and the  number  itself.  The

                          number 1 has only one factor, that is itself.

                    example 2. Find all the factors of 64.

                    Solution. We find the numbers whose product will be 64.

                           1 × 64 = 64 → 1 and 64 are factors of 64.

                           2 × 32 = 64 → 2 and 32 are factors of 64.                                        All possible
                                                                                                        combinations are
                           4 × 16 = 64 → 4 and 16 are factors of 64.                                      being tried out.

                            8 × 8 = 64 → 8 is a factor of 64.

                    Now, there are no other numbers which on multiplying give 64 as the product.

                    Thus, the factors of 64 are 1, 2, 4, 8, 16, 32 and 64.

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                                                                                           Multiples and Factors

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