Circle the highest common factor HCF of 10𝑥 cubed 𝑧 and 15𝑥 squared 𝑧 cubed. Five 𝑥 squared 𝑧, 150𝑥 to the fifth 𝑧 to the fourth, 30𝑥 to sixth 𝑧 to the
third, or five 𝑥𝑧.
So when finding the highest common factor, when looking at the numbers themselves
that in the 15, we need to break them down into their prime factors.
And to be prime means that it has exactly two factors: itself and one. So 10 could be written as two times five, where two is prime and five is prime. So that’s completely broken down. And for 15, we could rewrite that as three times five. Three is prime and five is prime.
Now, for the variables, it’s not called prime factorization. We just break it down into its single factors and see how many they have in
common. So 𝑥 cubed is the same as 𝑥 times 𝑥 times 𝑥, three 𝑥s. And then, 𝑧 is just simply 𝑧 or 𝑧 to the first power.
Now, for 𝑥 squared, for our next expression, it can be written as 𝑥 times 𝑥. And then, 𝑧 cubed can be rewritten as 𝑧 times 𝑧 times 𝑧. So now that both numbers are completely broken down, we want to circle the highest
And we have four to choose from. So let’s look at these.
For numbers, they both have five. They both have 𝑥s. And the greatest number of 𝑥s that they both have would be two. So we would write that as 𝑥 squared. And the highest number of 𝑧s that they both have would just be 𝑧.
Therefore, the highest common factor between them would be five 𝑥 squared 𝑧. So this will be our final answer.