What is the highest common factor of 24 and 90?

To find out the highest common factor of two different numbers, first of all we write each number as a product of its primes. 

Here we have

24 = 2 x 2 x 2 x 3 = 2^3 x 3^1

180 = 2 x 2 x 3 x 3 x 5 = 2^2 x 3^2 x 5^1

Now comes the important part. We compare the the prime factors of each number, and take the lower power of the two. So for example, we take out the 2^2 (2 squared) from the 180, because that power is smaller that the one for 24 (where we have 2^3). 

Similarly, we notice that 3^1 is the smaller power of the two cases (we have 3^2 for 180), and so we take out 3^1 (i.e. 3).

Finally, we see that the two numbers share no other prime factors, so now we simply multiply what we have taken out, and that it the highest common factor. i.e.

hcf(24, 180) = 2^2 x 3 = 4 x 3 = 12

And there's our answer. 

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