Raymond writes down a pair of numbers. One of these is a prime number, and the other is a factor of 12. He multiplies his prime number by three. He then adds the factor of 12. Raymond’s final answer is between 40 and 60. Find one of the possible number pairs that Raymond could have originally written down.
So the first thing I’m going to do to actually solve this problem is write down the primes. And I’m gonna write down the primes that when multiplied by three are actually going to be still less than 60 because then what we need to do is actually add a factor of 12 to them to make between 40 and 60. So before we can actually write these down, so write our prime numbers down, what we need to do is actually remember what they are. So what is a prime number? Well a prime number is a number that has exactly two factors. You may have heard another definition, which is that a prime number is a number that can be divided by itself and one. Well the reason that this doesn’t work is because of the number one.
The number one isn’t a prime number, however the number one can be divided by itself because one divided by one gives us an answer of one, so there’s no remainder, and it can also be divided by one because itself is one. However, it hasn’t got exactly two factors. So that’s how we can actually distinguish against one being a prime number. So okay, so now we know what prime numbers are. Let’s write down the ones that will satisfy the requirements for this question. So first of all we have two. Two is a special prime because it’s the only even prime number. So we’ve got two multiplied by three because we have to multiply the prime number by three because that’s what Raymond did. That give us six. Then we have three multiplied by three, which is nine; five multiplied by three, which is 15; seven multiplied by three, which is 21; 11 multiplied by three which is 33, 13 multiplied by three which is 39; 17 multiplied by three, which is 51; and our final one is 19 multiplied by three, which is 57.
And we stop there because 20 isn’t a prime number because it can be divided by two, 10, 20, five, four. So it hasn’t got exactly two factors. So therefore, everything else is gonna be above 20. And if you multiply a number above 20 by three, it will be over 60. And we’re only interested in numbers between 40 and 60. So now what we’re gonna do is actually work out the factors of 12. Our first pair of factors are one and 12, and that’s because one multiplied by 12 is equal to 12. And a factor of a number is a number that can actually multiply by another number to make the number that we’re talking about, which is in this case 12. Our next pair of factors is two and six because two multiplied by six is 12. And then we have our final pair of factors three and four, and that’s because three multiplied by four is equal to 12. We know it’s the final pair because the next number would actually be five, and five doesn’t go into 12 because all multiples of five end in a five or a zero.
Well now we have our primes and our factors of twelve, what we can actually do is actually rule out a section of our primes. Well we can rule out this section here because actually it says that Raymond’s final answer is between 40 and 60. So what we need to do is actually add our answer to our prime multiplied by three and one of our factors of 12. Well the highest factor of 12 is 12. So therefore, if we subtract 12 away from 40, which is our lower bound for Raymond’s final answer, we get 28. And therefore 28 must be the minimum value of our prime number multiplied by three. And as you can see the ones that we’ve crossed out are all less than 28, cause it’s 21, 15, nine and six. So now what we need to do is actually add the four values we have got left, which is 33, 39, 51, and 57 with one of our factors of 12.
And what we’re looking for is actually a final answer between 40 and 60. Well the question asked us to only find one of the possible number pairs that Raymond could’ve originally written down. So we’re gonna start with the first number we’ve got, which is 33. And that’s 11 multiplied by three, which gives us 33. We can see that if we added that to 12, then we’re gonna get an answer of 45. And 45 is actually between 40 and 60. So therefore, we can actually write down a pair of numbers that you could have actually chosen. Be careful not to fall into the trap to write down 33 and 12, because 33 wouldn’t have been an original number, because what it says is Raymond writes down a pair of numbers, one of these is a prime number and the other is a factor of 12. 33 is just a result of what happens when he actually multiplies this prime number by three. So therefore the two numbers that he would have written down are 11 and 12.
11 is the prime and 12 is the factor of 12. And these work because actually if you multiply 11 by three, you get 33 as discussed, and 33 add 12 would give us 45. So we’ve now solved the problem and given one of the possible number pairs that Raymond could have originally written down. It is worth noting there are a number of other different pairs that you could have had. And we could’ve found these by actually writing down the other numbers we had and adding them to the factors of 12. So we had 39, 51, and 57. And you can actually see that there’s actually a number of different combinations we could have got from these to make a number between 40 and 60 if we added them to one of our factors of 12.