A square number and a prime number have a total of 22. What are the two numbers?
In this problem, we’re told that the total of a square number and a prime number is 22. To work out what the two numbers are, we need to remember what a square number is and what a prime number is. A square number is formed by multiplying a number by itself. Let’s find as many square numbers as we can that are less than 22.
One squared has a value of one multiplied by one, which is one. So one is a square number. Two squared is the same as two multiplied by two. So the next square number is four. Three squared equals nine. And four squared equals 16. Five times five is 25; so that’s too large. The only square numbers that could feature in our calculation are: one, four, nine, or 16.
But what are the pairs that go with these numbers to make 22? One plus 21, four plus 18, nine plus 13, and 16 plus six. Now we need to ask ourselves, which of these second numbers are prime? What is a prime number? A prime number is a number with exactly two factors, one — well, all whole numbers have a factor of one — and itself. Let’s go through the factors of each of our numbers to see which one is prime. The factor pairs that make 21 are one times 21 and also three times seven. 21 has four factors; so we know it’s not prime. The pair of numbers we’re looking for can’t be these two.
How many factors does 18 have? The factor pairs that make 18 are one and 18, two and nine, and also three and six. 18 has six factors. So we know that’s not prime either. There’s only one multiplication that makes 13. And that’s one times 13. 13 is a prime number. It has exactly two factors, one and itself, which is 13. So it looks like we’ve found our square number and prime number. Let’s just check that six isn’t a prime number. Six has four factors: one, two, three, and six. And so we know it’s not prime. The pair of numbers are nine and 13.
We started off by reminding ourselves what a square number is. And then finding all the square numbers that we could add another number to, to get to twenty-two. We then worked out what the pairs were, what number we needed to add to the square numbers to make 22. And so we needed to find out which one of these was prime. We reminded ourselves what a prime number was. And then we went through each number to find out which one only had two factors, one and itself. A square number and a prime number have a total of 22. The square number is nine and the prime number is 13. Nine plus 13 equals 22.