Page 47 - Maths Class 06
P. 47

Lowest Common Multiple (LCM)
            The Lowest Common Multiple (LCM) of two or more numbers is the smallest number which is a multiple
            of each of the given numbers.
            Consider the numbers 8 and 12.

                   Multiples of 8 =  8, 16, 24, 32, 40, 48,…
                   Multiples of 12 = 12, 24, 36, 48, 60,…

            Common multiples of 8 and 12 are 24, 48,…
            The smallest of the common multiples of 8 and 12 is 24.
            Thus, LCM of 8 and 12 is 24.

            In order to find the LCM of two or more numbers, we explain the procedure below through examples:
            EX AM PLE 1.  Find the LCM of 24 and 36.

            SO LU TION :  The prime factorizations of 24 and 36 are
                                                24 =  2 2 2 3´ ´ ´
                                                36 =  2 2 3 3´ ´ ´
                          In these prime factorizations,

                          2 occurs at most three times (in 24)
                          3 occurs at most two times (in 36)

                          So, the LCM of 24 and 36 is 2 2 2 3 3´ ´ ´ ´ , i e. . 72.
            EX AM PLE 2.  Find the LCM of 15, 60 and 105.
            SO LU TION :  The prime factorizations of 15, 60 and 105 are

                                                15 =  3 5´
                                                60 =  2 2 3 5´ ´ ´

                                                105 = 3 5 7´ ´ .
                          In these prime factorizations,
                          2 occurs at most two times (in 60)

                          3 and 5 each occurs at most one time (in 15, 60 and 105)
                          7 occurs at most one time (in 105)
                          So, the LCM of 15, 60 and 105 is 2 2 3 5 7´ ´ ´ ´ , i e. . 420.

            Some Prop er ties of HCF and LCM
              1.   The HCF   of given numbers is not greater than any of     the given numbers.

              2.   The HCF of two co-primes is 1.
              3.   The LCM of given numbers      is not less than any of the given numbers.

              4.   The LCM of two co-primes is equal to their product.
              5.   The HCF of given numbers is always a factor of LCM.
              6.   If a  and b  are two given numbers such that a  is a factor of b, then their HCF is a  and LCM is b.

              7.   Product of two numbers =     Product of their  HCF and LCM.

                                                                                                    47
   42   43   44   45   46   47   48   49   50   51   52