Freema thinks of three different prime numbers. When she adds the three numbers together, the result is an even square number less than 30. What three prime numbers could she be thinking of?
The first thing we need to look at in this question is actually prime numbers. So what is a prime number? Well, a prime number is a number that has exactly two factors. We use this definition because actually it includes exactly what a prime number is.
There’s another definition you might hear, which is a prime number is a number that can only be divided by itself and one. However, the number one is not a prime. And the number one can be divided by itself without a remainder, because one divided by one is one, and the number one can also be divided by one without a remainder. That’s because itself is one. But one is not a prime. So that’s why we say a prime number must have exactly two factors and no more and no less. And that’s because one has only one factor so therefore would not count in this definition.
So what I’ve done first is actually listed the prime numbers we know up to 30. So we’ve got two, three, five, seven, 11, 13, 17, 19, 23, and 29. Well, we can instantly remove 29. And that’s because it says that when Freema adds three numbers together, the result is an even number less than 30. Well, actually, 29 add two and three, which would be the other two smallest prime numbers, would give us a number greater than 30. So now we’ve ruled one of those out, let’s have a look at the numbers less than 30 that are square numbers.
So let’s quickly remind ourselves what a square number is. A square number is the result of multiplying a number by itself. So, for example, two multiplied by two is four, so four is a square number. So we’re gonna say that two squared is equal to four. So as we said, four is a square number.
Now we’re only interested in the ones that are actually even numbers. So therefore, the only two we’re interested in are four and 16. That’s cause one squared is one, and that’s an odd number; three squared is nine, again an odd number; five squared is 25, which is an odd number; and then six squared is 36. But 36 is actually greater than 30. So therefore, we’re not interested in that number.
Okay, great, so now what we need to do is actually think of three different prime numbers that when added together will give us either four or 16. Well, we can also rule out another one of our square numbers, and that’s four. That’s because the smallest actual sum that we could get would be when we added two, three, and five. And two plus three plus five is actually equal to 10, and 10 is greater than four. So therefore, four cannot be the number that we’re looking for either.
So therefore, 16 is the result that we’re looking for when we add together our three prime numbers. So therefore, we can instantly rule out three more of our prime numbers, 23, 19, and 17, cause these are all already greater than 16. We can also rule out 13, and that’s because the smallest sum that can be formed with 13 is 13 plus two, which is 15, plus three, which is 18. And 18 is also greater than 16.
So therefore, we’re gonna have a look at 11. And what we’re gonna do is actually try adding the two smallest primes to 11. And we’re doing this because 11 is actually the prime number closest to 16 that’s left, so we want to try this one first.
Well, if we do 11 add two add three, this is gonna be equal to 16. That’s because 11 add two is 13, add three is 16. So therefore, we can say that the three different prime numbers that Freema is thinking of are two, three, and 11. So we’ve got the final answer, but it’s always worth checking to see if actually there were any others.
Well, we can’t add 11 to any other numbers, and that’s because if we had 11 add five, then that would already be 16 before adding a third. So let’s have a look at seven. Well, if we tried seven, then we’d have to add it to five and three. Well, seven add three add five is gonna give us seven add three is 10, add five is 15. And 15 is less than 16. So therefore, definitely the only three numbers that Freema could have actually chosen are two, three, and 11.