Question

2. It is given that P = (x:x is an integer such that 0<x< 13).
Q = (x:x is a prime number less than 13)
and R = (x:x is a positive integer not more than 13).
(1) List all the elements of P and of Q.​

Answer 1

Answer:

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By medv– Topcoder Member

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In addition to being a Topcoder member, medv is a lecturer in Kiev National University’s cybernetics faculty.

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.

Exercise 1. Sequence an is defined recursively:

Prove that ai and aj, i ¹ j are relatively prime.

Hint: Prove that an+1 = a1a2…an + 1 and use Euclid’s theorem.

Exercise 2. Ferma numbers Fn (n ≥ 0) are positive integers of the form

Prove that Fi and Fj, i ≠ j are relatively prime.

Hint: Prove that Fn +1 = F0F1F2…Fn + 2 and use Euclid’s theorem.

Dirichlet’s theorem about arithmetic progressions:

For any two positive coprime integers a and b there are infinitely many primes of the form a + n*b, where n > 0.

Trial division:

Trial division is the simplest of all factorization techniques. It represents a brute-force method, in which we are trying to divide n by every number i not greater than the square root of n. (Why don’t we need to test values larger than the square root of n?) The procedure factor prints the factorization of number n. The factors will be printed in a line, separated with one space. The number n can contain no more than one factor, greater than n.

Answer 2

Step-by-step explanation:

we have to find integers greater than 0 and less than 13

i.e. P={1,2,3,4,5,6,7,8,9,10,11,12}

here it is clear that the values of x are greater than 0 and less than 13 ,and all the values are integers

now,

Q={2,3,5,7,11}. because in the set Q we have to write all the prime no.less than 13