Question

Define 'prime number.' Write down the last three digite of your social 5SN216 security number and factor it into a product of prime numbers. (b) Prove Euclid's theorem that there are infinitely many prime numbers.

Answer 1

A prime number is a natural number greater than 1 that can't be formed by multiplying two smaller natural numbers. Euclid's theorem affirms the existence of an infinite number of prime numbers, proven by contradiction by assuming a finite list and showing it cannot account for all primes.

Step-by-step explanation:

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, the numbers 2, 3, 5, and 7 are prime numbers because they can only be divided evenly by 1 and the number itself.

Euclid's Theorem on Prime Numbers

Euclid's theorem states that there are infinitely many prime numbers. This can be proved by contradiction: Assume there is a finite list of prime numbers. We can construct a new number by multiplying all the primes together and adding 1. This new number is not divisible by any of the primes on our list.

Thus, either it is a new prime or it has prime factors not on our list, which contradicts the assumption that we had listed all the primes. Therefore, there must be infinitely many primes.