Write 408 as a product of prime factors.
Right, let’s just look at the words used in this question first of all. The word product means to multiply. The prime factors of a number are numbers that divide exactly into that number and are themselves prime numbers. So they have no factors other than one and themselves. To write a number as the product of its prime factors, we can draw a factor tree.
We start by finding any factor pair of the number that we’re given. And these numbers do not have to be prime. So first of all, I notice that 408 can be divided by four. 408 is equal to four multiplied by 102. So this is the factor pair that I’m going to start off with. Now, neither of these numbers are prime numbers. So we have to continue our factor tree.
Four can be written as two multiplied by two. Now, two is a prime number. In fact, it’s the only even prime number. So we can stop this branch of the factor tree here. We circle the twos to show that they’re prime numbers. Now, we need to carry on with the other branch of our factor tree.
102 is an even number and half of 102 is 51. So we can write 102 as two multiplied by 51. Remember two is a prime number. So we can circle it. Now, we need to decide whether or not 51 is a prime number.
Now, at first glance, you might think, “yes, it is” as you probably don’t recognize it for many of your times tables. However, 51 is actually in the three times table. And there’s an easy way to check this. To see if a number can be divided by three, you can add its digits together. And if the answer can be divided by three with no remainder, then so can the original number.
So for a 51, the sum of its digits five and one is six. And six can be divided by three. It gives two. So if six is divisible by three, then so is the original number 51. Now, this only works for checking whether a number is divisible by three. And actually, there is a similar method for checking whether a number is divisible by nine. But you can’t use a method like this for checking if a number is divisible by another number such as four, five, and so on.
To work out what 51 divided by three is, we can use a short division or bus stop method. Three goes into five once with a remainder of two and three goes into 21 seven times with no remainder. So 51 is equal to three multiplied by 17. Now, three and 17 are both prime numbers. So we can circle them and our factor tree stops here.
All the branches in our factor tree now end in prime numbers. So we finished the factor tree. Now, we found all of the prime factors of 408. But remember we need to write 408 as the product of its prime factors. So we collect all of the factors from our factor tree and write them as a product: two multiplied by two multiplied by two multiplied by three multiplied by 17.
It is usual to list the factors in order from the smallest to the largest. Now, we can actually simplify this answer slightly as two multiplied by two multiplied by two can be written as two cubed or two to the power of three. So we can write our answer as two cubed multiplied by three multiplied by 17.
This is called giving your answer in index form as another word for power is index. Now, it’s just worth mentioning you can actually start off with any pair of factors of the original number and still get the same result. If we hadn’t started with four and 102 as our first factor pair, we could’ve instead used two and 204, for example.
You could draw out the factor tree starting with this pair of factors and confirm that you always end up with the same list of factors at the end. They may initially be in a different order. But you’ll always have the same list.
As a product of prime factors, 408 is equal to two cubed multiplied by three multiplied by 17.