Write 306 as a product of its prime factors.
To write a number as the product of its prime factors, we must first work out what its prime factors are. The prime factors of a number are its factors which have exactly two factors themselves. For example, five is prime because its only factors are one and five. So it has exactly two factors, whereas six isn’t prime because it can be written as one multiplied by six or two multiplied by three. It has four factors.
Notice that the number one is not a prime number because it can only be written as one multiplied by one. It has one factor rather than two. It’s a common mistake though to think that one is a prime number. But if you remember that the definition of a prime number is that it has exactly two factors, then you’ll see why one can’t be a prime number. To find the prime factors of any number, we can draw a factor tree. We start by breaking this number down into any factor pair.
Now there are a couple of different ways we could do this. I know that this number is even because it ends in a six. That’s an even number. But I can also spot fairly easily that it’s divisible by three because 300 is divisible by three. And this number is only six more than that. Six is also divisible by three. 300 divided by three is 100, and six divided by three is two. So this means that 306 divided by three is 102. Three is a prime number. So we circle it. And this branch of the factor tree stops here.
However, 102 isn’t. We know this because it ends in a two. It’s an even number, which means it can be divided by two. 102 divided by two is 51. Two is prime. So we circle it. Remember, two is the only even prime number. But what about 51? We know that 51 won’t be divisible by two because it’s an odd number. So let’s check the next number.
Is 51 divisible by three? In fact, there’s a quick check you can use to see whether any number is divisible by three. What we do is we add its digits together. So for 51, five plus one is equal to six. If the number you get when you add the digits together is divisible by three, then this means that the original number is also divisible by three. So in this case, six is divisible by three, which means 51 is also divisible by three. This trick is really useful. But it only works for three and actually nine as well. But you can’t check divisibility in general using this method. For example, you couldn’t add all the digits together and see that it was divisible by four to check whether a number was divisible by four itself.
To divide 51 by three, we can either do this mentally or we can use a bus stop division method. Three goes into five once, with a remainder of two. And three goes into 21 seven times exactly. So 51 divided by three is 17. Both three and 17 are prime numbers. So we can circle them, and our factor tree stops here.
It’s a good idea to be familiar with prime numbers up to about 30 so that you can recognise them easily in questions like these. But if you didn’t know that 17 was a prime number, you’d just have to check whether it could be divided by numbers like two, three, and four. You can rule all the even numbers out straight away as 17 is an odd number.
So we completed our factor tree. And now we need to write 306 as the product of its prime factors. We collect up all of the circled numbers. That’s the prime factors on our factor tree. And we usually write them in ascending order. That’s from smallest to largest. We have a two, two threes, and one factor of 17. So 306 is equal to two times three times three times 17. You can check this on your calculator in this question.
Now this question hasn’t specifically asked for this. But often in questions like this, we’re asked to give our answer in a particular form. It’s called index form. This means that if any factors appear more than once — so in our case, we have three times three — we write this using a power or index. So three times three could be written as three squared.
We can therefore write our answer as two times three squared times 17. And we’ve written 306 as the product of its prime factors. And in this case, we’ve given our answer in index form.