The Greek mathematician, Euclid, demonstrated circa 300 BC that given any finite set of prime numbers, we can always produce another prime number not in that given set. One obvious conclusion is that there must be infinitely many prime numbers. This statement is known as Euclid’s theorem and we will examine his method here. Suppose that we have a set of prime numbers, {p1, p2, …, pn}. Using this set of prime numbers, we compute a new number as follows: x = p1 × p2 × … × pn + 1. Each of the prime factors of x is a “new” prime number and does not appear in the original given set. For example, suppose that we have the following set of prime numbers, {5, 11}. Then, x = 5 × 11 + 1 = 56. The prime factors of 56 are 2 and 7 since 56 = 2 × 2 × 2 × 7. Note that the prime numbers 2 and 7 do not appear in the original set.

- 1. If you are given the set of prime numbers {13, 47} which new prime numbers are produced using the method described above?
- 2. If you are given the set of prime numbers {5, 11, 23} which new prime numbers are produced using the method described above?
- 3. If you are given the set of prime numbers {7, 17, 31} which new prime numbers are produced using the method described above?

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