Question

What is the GCF of 225 and 315?

Answer 1

`45`

Explanation:

List the factors of `225` and `315` and find the highest one that both share.

You could also do it by realising that the GCF of `225` and `315` also has to be a factor of the difference between the two of them. In other words, the factor has to go into `90`, because otherwise you would have no way of linking `225` to `315` with whole numbers. And the greatest factor of one (even) number is half of it, which means `90/2 = 45` is a great suspect for the GCF. It's as good a starting point as any, and in this case turns out to be right.

Answer 2

`GCF = 3xx3xx4=5 = 45`

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.

It is obvious that 5 is a common factor, but without some working we cannot decide on a bigger factor.
You might recognise `225 " as " 15^2`

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

`color(white)(xxxx) 225 = 3xx3xx5xx5`
`color(white)(xxxx) 315 = 3xx3xx5color(white)(xxx)xx7`

`GCF =color(white)(xxx)3xx3xx5 color(white)(xxxxxx)=45`

From this it is very clear that the common factors are 3x3 x5.

If we needed the LCM as well 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 = 3xx3xx5xx5xx7 = 1575`