Find the highest common factor,
(HCF) of 21, 56, and 112.
To do this, we can create factor
trees. So we will be breaking down these
numbers into their prime factors. To be prime means to have exactly
two factors. For example, two is prime. Two has factors of two and one
because two times one is two. Now, one is not prime because one’s
factors are one and one for one times one. And there’re not exactly two
factors; there’s only exactly one factor. So one is not prime.
So let’s begin by breaking these
numbers down. 21 is seven times three. Now, seven is prime because it only
has two factors: one and seven and three is prime. So we broke down 21 into its prime
Now, for 56, 56 is eight times
seven. Now, eight is not prime; it has
more than two factors. So let’s break down eight. Eight is four times two. Four is not prime; four has factors
of two and two. And two is prime because two only
has two factors: itself and one. So all of the twos are prime and
the seven is prime. So we’ve broken this down into its
Now for 112, 112 is equal to two
times 56. And we know exactly how 56 breaks
down because we just did that on the left. 56 is eight times seven. And eight breaks down into four
times two and then four breaks down into two times two. So all of these twos are prime and
the seven is prime. So we know that 21 is seven times
three. 56 is two times two times two which
is two cubed times seven. 112 would be two to the fourth
power times seven.
So what would be the highest common
factor between these three? It would be seven. Two to the third power and two to
the fourth power yes indeed are larger. However, 21 does not have a two to
any power. So we can’t use that. If it did, we would use the highest
power of two that all three of them would have.
But once again, the highest common
factor of 21, 56, and 112 would be seven.