Question

How do you find the greatest common factor of 9, 6, and 42?

Answer 1

Hint: To find the greatest common factor, also known as the highest common factor, one needs to identify all the numbers which can be factors of the numbers in question and then find which of them are common and multiply them to get the highest factor.

Complete step by step solution:
We have the numbers 9, 6, and 42. To proceed further, first, we need to break each number into its multiples. First, we shall start at 9.
$9 = 1 \times 3 \times 3$
Then moving on, we observe the multiples of 6.
$6 = 1 \times 2 \times 3$
Next, we observe 42.
$42 = 1 \times 2 \times 3 \times 7$
In all the three cases above, we find that 1 is the common factor so that will be our first common factor. Next, we find that 3 is also the common factor which makes it so that 3 is also the second common factor.
Apart from them, we find 2 as a common factor in both 6 and 42 but it is not present in 9 so it cannot be considered as a common factor. Other than that, we can find nothing common in the factors of the three numbers which are 9, 6, and 42 so we move forward to finding the highest common factor.
${\text{H}}{\text{.C}}{\text{.F}}{\text{. = 1}} \times {\text{3 = 3}}$
So, the highest common factor is 3.
A more traditional method of finding the highest common factor is
$\
  3\left| \!{\underline {\,
  {9,6,42} \,}} \right. \\
  \,\,\,\,\,3,2,14 \\
\ $
Here on the left side of the bar, we use the numbers to divide, and on the right side are the numbers whose HCF is to be found out. Then we multiply the numbers appearing on the left side to find the HCF. In this case, there is only one number which is 3 but if there were other common multiples, the division would have continued further until there was no common factor left.

Note:
This same method is used to find the least common multiple too where after finding the HCF, all the other left multiples would have been multiplied into the HCF to find the LCM.