Question

What is the GCF of 180, 108, and 75?

Answer 1

Greatest Common Factor is `3`.

Explanation:

Factors of `180` are `{1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180}`

Factors of `108` are `{1,2,3,4,6,9,12,18,27,36,54,108}`

Factors of `75` are `{1,3,5,15,25,75}`

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

Greatest Common Factor is `3`.

Answer 2

`GCF = 3`

Explanation:

In most cases we should be able to find the GCF fairly easily by just knowing the multiplication tables up to 12 x 12. Sometimes a bigger number might be included which we do not know well. This is just such a case.
Using factor trees mentally will allow you write all the prime factors.

(for example: `108 = 12 xx 9 = (4xx3)xx(3xx3) = 2xx2xx3xx3xx3`)

It is good to have a method available for cases when we cannot find the GCF by inspection.

In order to find the GCF (and the LCM) write each number as the product of its prime factors.

`color(white)(xxxx) 180 = 2xx2xx3xx3color(white)(xxx)xx5`
`color(white)(xxxx) 108 = 2xx2xx3xx3xx3`
`color(white)(xxxx)75 = color(white)(xxx..xx)3color(white)(xxx..x)xx5xx5`

`GCF =color(white)(xxxxxxxxx)3`

From this it is very clear that the only common factor is 3.
(I find this result surprising - I would thought it would be higher.)

If we needed the LCM it can be calculated easily from this format:
Include each column of factors, do not count factors that are in the same column twice.

`LCM = 2xx2xx3xx3xx3xx5xx5 = 2^2 xx 3^3 xx 5^2 = 2700`