Question

Find the smallest prime number that divides both of them or state that such number does not exist.

Answer 1

Answer:

Prime numbers and their properties were extensively studied by the ancient Greek mathematicians. Thousands of years later, we commonly use the different properties of integers that they discovered to solve problems. In this article we’ll review some definitions, well-known theorems, and number properties, and look at some problems associated with them.

A prime number is a positive integer, which is divisible on 1 and itself. The other integers, greater than 1, are composite. Coprime integers are a set of integers that have no common divisor other than 1 or -1.

The fundamental theorem of arithmetic:

Any positive integer can be divided in primes in essentially only one way. The phrase ‘essentially one way’ means that we do not consider the order of the factors important.

One is neither a prime nor composite number. One is not composite because it doesn’t have two distinct divisors. If one is prime, then number 6, for example, has two different representations as a product of prime numbers: 6 = 2 * 3 and 6 = 1 * 2 * 3. This would contradict the fundamental theorem of arithmetic.

Euclid’s theorem:

There is no largest prime number.

To prove this, let’s consider only n prime numbers: p1, p2, …, pn. But no prime pi divides the number

N = p1 * p2 * … * pn + 1,

so N cannot be composite. This contradicts the fact that the set of primes is finite.

Step-by-step explanation: