Here are five numbers: two, three, four, five, and six. Write each number on the correct cards. The number two has been written on the correct cards for you. And then we’re given three cards labelled prime numbers, factors of 12, and factors of 15.
There are two keywords in this problem that we need to understand if we’re going to get the correct answer. If we don’t know what they mean, we’re gonna find it difficult to answer the problem.
The first keyword or phrase is prime numbers. All of the numbers that we write on the first card have to be prime. We’ll go over what that means in a second. The second keyword appears on two of the cards. It’s the word “factors.” We need to find factors of 12 and factors of 15. So we’ve got three cards containing phrases that we need to understand. And what do we need to do to solve the problem?
Well, we’re given five numbers: two, three, four, five, and six. And we need to write those numbers whatever we need to on the cards. If a number belongs on more than one card, we need to write it more than once. And we’re given a big clue as the number two has already been written on the correct cards for us. So let’s remind ourselves what the word “prime numbers” and “factors” mean. And while we do so, we’ll have a think about why the number two belongs where it does.
Because a prime number is to do with the factors of a number, it might make sense to go over what the word “factors” means first. The factors of a number are numbers that can divide into that number without leaving any remainder. So the factors of 10, for example, are any numbers that we can divide 10 by and get a whole number answer. We can divide 10 by two, so two must be a factor of 10. We can also divide 10 by five and get a whole number answer. And so five must also be a factor of 10.
On the final two cards, we need to collect together all the factors of 12. We can divide 12 by two and get six. So two is a factor of 12. That’s why the number two has been written on there. And then the last card shows the factors of 15. Well, although we can divide 15 by two, it’s not going to give us a whole number answer. It doesn’t divide without a remainder. 15 isn’t in the two times table, so two is not a factor of 15.
Let’s come back to the phrase “prime numbers.” What’s a prime number? Prime numbers are numbers with only and exactly two factors: one and themselves. Some numbers have a few different factors. We can divide every whole number by one. And so one is a factor of every single whole number. And every number has itself as a factor. It can be divided by itself. But the number two, for example, can’t be divided by anything else. We can only divide it by one and itself or two. There are only two factors, and so the number two is a prime number.
Now we can start to solve the problem. Let’s use what we know about prime numbers and factors to work out where the numbers three, four, five, and six belong. Which numbers are prime? A good way to find the different factors that make up a number is to think about pairs of numbers that multiply together to make that number.
We can make the number three by multiplying together one and three. But there aren’t any other whole number multiplications that make three. The only factors of three are one and itself. Three is a prime number.
What about the number four? How can we make four? One times four is four. That’s our prime number multiplication, one and itself. But also two times two is four. So four also has two as a factor. Four is not a prime number. It has more than two factors.
What about the number five? One times five equals five. That’s one and itself. And there are no more multiplications involving whole numbers that make five. Five is a prime number. The only factors it has are one and itself.
Finally, how can we make six? One times six, but also two times three. The factors of six are one, two, three, and six. Six is not a prime number. It has four factors, not two.
Now let’s think about the second card. Which of our numbers are factors of 12? Let’s list a different multiplications that make 12 so that we can find each factor. One times 12 equals 12. Two times six equals 12. And three lots of four make 12 too. So the number 12 has six factors: one, two, three, four, six, and 12. The numbers that’s from our list are three, four, and six.
Finally, let’s list the factors of 15 so we can see which of our numbers belong on the final card. One times 15 equals 15. And three lots of five make 15. The two numbers then that appear on our cards are three and five.
So we used our knowledge of factors to help answer this question. And also we had to understand what a prime number was. If we ignore the number two because it was done for us, the prime numbers in our list were three and five. The factors of 12 were three, four, and six. And the factors of 15 were three and five.