LCM and HCF

(a) Write 176 as the product of it's prime factors. Give your answer in index form.

As soon as you see prime factors you should think factor tree

       176

        /  \

     2     88               the numbers in red are prime numbers, so, 2,2,2,2,11 are

           /    \              it's prime factors. We write it as 2⁴ x11 (which is index form)

        2       44           we write it like this because, 2⁴ x 11 = 176

                /    \

              2     22

                     /     \

                  2       11

(b) Hence, find the HCF of 176 and 100.

this just means find the prime factors of 100, then multiply whatever prime factors that are the same as 176's prime factors.

        100

        /     \

      2      50                    The prime factors of 100 are 2, 2, 5, 5

             /     \                  The prime factors of 176 are 2, 2, 2, 2, 11

          5        10              We cross out the ones that do not match

                    /    \            So, the HCF would be 2 x 2 x 2 x 2 or 2⁴, 2⁴ = 8

                  2       5

So the HCF of 176 and 100 is 8

(a) Find the LCM of 24 and 56.

       24                              56

       /    \                           /     \

     2     12      2,2,2,3     2       28           2,2,2,7

           /    \                            /    \

         2     6                         2     14

              /     \                             /    \

             2      3                          2      7

so we multiply the prime factors, 2 x 2 x 2 x 3 x 7 (we don't need to put 2 x 2 x 2 x 2 x 2 x 2, because they are common factors we only put one set of 2 x 2 x 2)

This gives us 168. so, the LCM of 24 and 56 is 168.