What is the greatest common factor of 42, 63, and 105?

Factors of `42` are `{1,2,3,6,7,14,21,42}`

Factors of `63` are `{1,3,7,9,21,63}`

Factors of `105` are `{1,3,5,7,15,21,35,105}`

Common factors are just `{1,3,7,21}` and

Greatest Common Factor is `21`.

Writing each number as the product of its prime factors is a quick way of finding the HCF and LCM of any number of values.

`" " 42 = 2xxcolor(blue)(3xx" "7)`
`" "63 = " "color(blue)(3)xx3xxcolor(blue)(7)`
`" "ul(105= " "color(blue)(3 xx" "7) xx 5)`
`HCF =color(white)(xxx)color(blue)(3xx" "7)" "= 21`

`42,63 and 105` all have the factors `3 and 7` in common.

Therefore the `HCF =21`

What is the greatest common factor (GCF)?
That is the largest number that will divide into all those given.
To find it, the smallest prime numbers should be divided into each one. Prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19.

Looking at `42, 63, 105,` we can see that the first is divisible by `2`, the second by `3` and the third by `5`:

`42/2=21` and `63/3=21` and `105/5=21`

Each division results in the same number, so that is the GCF.