Number Theory

Lesson

A factor tree is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime. Once we completely break down a number into its prime factors, we can use them to write it in index notation.

Index notation is a short way of writing a number being multiplied by itself several times. For example, $5\times5\times5$5×5×5 can be written as $5^3$53.

How can we write $8$8 in index notation, as a product of its prime factors?

So we can write $8$8 as $2\times2\times2$2×2×2. When we write this in index notation, it would be $2^3$23.

Let's look at another example. How would we write $81$81 in index notation?

So we can break $81$81 down into $4$4 sets of multiplication by $3$3, so in index notation, we would write this as $3^4$34.

Sometimes there is more than $1$1 prime factor. Let's look at $36$36 as an example:

$36$36 can be broken down so in index notation we would write this as $2^2\times3^2$22×32 .

Consider the number $225$225.

First, write $225$225 as a product of prime factors. Give your answer in expanded form.

Now list all factors of $225$225, separating factors with a comma.

We want to find all factors of $30$30 by using a factor tree.

First, write $30$30 as a product of prime factors.

Now list all factors of $30$30, separating factors with a comma.

Write $72$72 as a product of prime factors in index form.