1.1.01. Natural numbers

Dictionaries

Natural numbers - keywords

Sinopisis

  1. The set of natural nubers and sympolic notation $N=\{ 1, 2, 3, ...\}$.
  2. What does it mean that a natural number $d$ is a divisor of a natural number $n$?
  3. What does it mean that a natural number $n$ is a multiple of a natural number $d$?
  4. Why, instead of divisor you can say factor?
  5. What does the abbreviation $d|n$ mean?
      Istead you can say:
    • $d$ is a divisor of $n$
    • $d$ is a factor of $n$
    • $n$ is divisible by $d$
    • $n$ is amultiple of $d$
  6. Even numbers $0, 2, 4, 6, 8, ... $ nad odd numbers $1, 3, 5, ...$.
  7. Division with remainder $n=k\times d+r$
  8. Prime numbers: 2, 3, 5, 7, 11, 13, ...
  9. Euklides's Theorem ($\infty $)

    There are infinitely many prime numbers.
    Proof
    (from Euclid's theorem - Wikipedia)
    Consider any finite list of prime numbers $p_1$, $p_2$, ..., $p_n$.
    It will be shown that at least one additional prime number not in this list exists.

    Let $P$ be the product of all the prime numbers in the list: $P = p_1\cdot p_2\cdot ...\cdot p_n$.
    Let $q = P + 1$.
    Then $q$ is either prime or not:

    • If $q$ is prime, then there is at least one more prime that is not in the list.
    • If $q$ is not prime, then some prime factor $p$ divides $q$. If this factor $p$ were in our list, then it would divide $P$ (since $P$ is the product of every number in the list); but $p$ divides $P + 1 = q$. If $p$ divides $P$ and $q$, then $p$ would have to divide the difference of the two numbers, which is $(P + 1) − P$ or just $1$. Since no prime number divides $1$, $p$ cannot be on the list.

    This means that at least one more prime number exists beyond those in the list.
    This proves that for every finite list of prime numbers there is a prime number not in the list, and therefore there must be infinitely many prime numbers.

  10. The sieve of Eratosthenes
  11. Composite numbers: 4, 6, 8, 9, 10, 12, 14, ...
  12. Unique Prime Factorization
  13. GCD and LCM
  14. Divisibility tests

You can do it

Exercise 1. Factorize number 504.
Exercise 2. Factorize number 2250.
Exercise 3. Find GCF(2250,504).
Exercise 4. Find LCM(2250,504).
Exercise 5. Determine whether 7168 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.