Fundamental Theorem of Arithmetic

Description

This lecture explains the concept of prime numbers, the fundamental theorem of arithmetic, and how any natural number can be uniquely decomposed into a product of prime factors. Through an example of decomposing 362,880, the instructor demonstrates the process of finding prime factors and calculating the greatest common divisor and least common multiple. The fundamental theorem of arithmetic has practical implications for calculations, simplifying the process of finding common divisors and multiples of large numbers.

Official source

https://mediaspace.epfl.ch/media/0_vhynv9jl

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Related concepts (61)

Greatest common divisor

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted . For example, the GCD of 8 and 12 is 4, that is, . In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc.

Least common multiple

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a, 0) as 0 for all a, since 0 is the only common multiple of a and 0.

Divisor

In mathematics, a divisor of an integer , also called a factor of , is an integer that may be multiplied by some integer to produce . In this case, one also says that is a multiple of An integer is divisible or evenly divisible by another integer if is a divisor of ; this implies dividing by leaves no remainder. An integer n is divisible by a nonzero integer m if there exists an integer k such that . This is written as Other ways of saying the same thing are that m divides n, m is a divisor of n, m is a factor of n, and n is a multiple of m.

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.

Factorization

In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as whenever is not zero.