Question

What is the GCF of 210 and 252?

Answer 1

`42`

Explanation:

One way of finding the GCF of two numbers is as follows:

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Divide the larger number by the smaller to give a quotient and remainder.

If the remainder is zero then the smaller number is the GCF.

Otherwise, repeat with the smaller number and the remainder.

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In our example:

`252 / 210 = 1` with remainder `42`

`210 / 42 = 5` with remainder `0`

So the GCF is `42`

Answer 2

`GCF = 42`

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: `210 = 10xx21 = 2xx5xx3xx7`)

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) 210 = 2color(white)(xxx)xx3xxcolor(white)(x)xx5xx7`
`color(white)(xxxx) 252 = 2xx2xx3xx3color(white)(xxx)xx7`

`GCF =color(white)(xxx)2color(white)(xxx)xx3color(white)(xxxx.x)xx7 =42`

From this it is very clear that the common factor is 42

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.