OU on the BBC: The Code - Prima Pizzeria hints and tips

1. Prima Pizzeria is about prime numbers. A prime number is a whole number greater than 1 that is divisible only by itself and 1. For example, 7 is a prime number because it is divisible only by 7 and 1, but 9 is not a prime number because it is divisible by 3. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
2. The only even prime number is 2. That’s because every even number other than 2 is divisible by 2!
3. Prime numbers are the building blocks of all whole numbers. That’s because every whole number greater than 1 can be obtained by multiplying prime numbers together. For instance, 60 = 2 × 2 × 3 × 5.
4. Whole numbers greater than 1 that are not prime numbers are called composite numbers. In Prima Pizzeria you must spread a composite number of toppings evenly over a pizza with a prime number of slices. So given a batch of 60 toppings you must choose a pizza with either 2, 3, or 5 slices, because 2, 3, and 5 are the only prime numbers that divide 60
5. Prima Pizzeria tests your dividing skills! For instance, given a batch of 8 toppings, a pizza with 3 slices is no use, because 3 doesn’t divide 8. Instead you need a pizza with only 2 slices, because 2 is the only prime number that divides 8.

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6. Suppose we have a number (which we call A) that is divisible by another number (which we call B). Then B is said to be a factor of A. The numbers 2, 3, and 5 are all factors of 60. There are other factors of 60 too; such as 4, 15, and 20. Prima Pizzeria is about searching for prime number factors of composite numbers.

7. To test whether a number is divisible by 3, add the digits of that number together, and determine whether the resulting number is divisible by 3. For example, consider the number 513. Add 5, 1, and 3 together to obtain 9, which is divisible by 3. So 513 is also divisible by 3, and in fact 513 = 171 × 3. On the other hand, consider the number 514. Add 5, 1, and 4 together to obtain 10, which is not divisible by 3. So 514 is not divisible by 3.

8. You can work out all the prime numbers less than 100 as follows. Create a ten by ten grid containing the numbers 1 to 100, in order. Cross off the number 1. Now cross off all multiples of 2 other than 2 itself, namely 4, 6, 8, …, 100. Next cross off all multiples of 3 other than 3 itself, namely 6, 9, 12, …, 99. Repeat this for the prime numbers 5 and 7. The remaining numbers that haven’t been crossed off are the prime numbers. This method for identifying prime numbers is known as the Sieve of Eratosthenes.

9. Every composite number has a prime factor that is no greater than the square root of the composite number. For example, consider the composite number 100. The square root of 100 is 10, and 5 is a prime factor of 100 which is less than 10.

10. The ancient Greek mathematician Euclid proved that there are infinitely many prime numbers. This means if you tried to list the prime numbers then your list would never end. Euclid established this fact using a proof by contradiction: from the assumption that there are only finitely many prime numbers, he deduced an impossible conclusion. This meant his assumption was false, and on the contrary there are indeed infinitely many prime numbers.