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Maths Query > Unit > Arithmetic > Arithmetic Theorems

# Fundamental Theorem of Arithmetic with Applications

## Theorem Statement

Every number can be expressed as product of primes and this factorisation is unique, apart from the order in which the prime factors occur.

Example

Consider a number 45.
The factors of 45 are 3, 3 and 5.
45 = 3 × 3 × 5
Here, the factors 3, 3, and 5 are prime numbers.
So, it shows the factors of a number are primes which are 3, 3, and 5.
Also, the number 45 can be expressed as product of primes.

Note

Every number in maths can either be a prime or a composite number except 0 and 1 because 0 and 1 are neither a prime number nor a composite number.

## Prime factorization is always unique

Fundamental theorem of arithmetic states that prime factorization is always unique irrespective of order. Let’s see how prime factorization is unique with an example to factorize 48 in two different ways with order of factors different, but the prime factors remain the same.

Example

From the above figure, the order of prime factors are 2, 2, 2, 2, 3
Product of primes = 2 x 2 x 2 x 2 x 3 = 48

From the above figure, the order of prime factors are 2, 3, 2, 2, 2
Product of primes = 2 x 3 x 2 x 2 x 2 = 48

So, in both ways the prime factors and their product remains the same, though they differ in the occurrence of order of prime factors.

## Applications of fundamental theorem of arithmetic

The applications of fundamental theorem of arithmetic can be seen while finding out highest common factor and least common multiple of numbers.

The first step to find out HCF and LCM of given numbers is to convert the given numbers into their factors. Such factors of the numbers always come out to be the prime numbers, where fundamental theorem of arithmetic applies.

### 1. GCD or HCF

The factors of given numbers are found out which always comes out to be the prime numbers as per fundamental theorem of arithmetic, which states every number can be expressed as product of primes.
In next step, the lowest power of common factors are taken and their product is calculated, which is the GCD of given numbers.
GCD of two numbers a and b is, gcd(a, b) = product of least powers of common prime factors

Example

Find HCF of 12 and 15.
Prime factorization of 12 = 2 × 2 × 3
(as per fundamental theorem of arithmetic, every number can be expressed as product of primes, so here 12 can be expressed as product of prime numbers 2 and 3)
Prime factorization of 15 = 3 × 5
(similarly, here, fundamental theorem of arithmetic applies, so 15 can be expressed as product of prime numbers 3 and 5)
Common factors of 12 and 15 with least power = 31
∴ HCF of 12 and 15 = 31 = 3

### 2. LCM

The factors of given numbers are found out which always comes out to be again the prime numbers as per fundamental theorem of arithmetic.
In next step, the highest power of each factor from the factors of given numbers are taken and their product is calculated, which is the LCM of given numbers. lcm(a, b) = product of greatest powers of each prime factors

Example

Find LCM of 12 and 15.
Prime factorization of 12 = 2 × 2 × 3
(as per fundamental theorem of arithmetic, here 12 can be expressed as product of prime numbers 2 and 3)
= 22 × 31
Prime factorization of 15 = 3 × 5
(similarly, here, fundamental theorem of arithmetic applies, so 15 can be expressed as product of prime numbers 3 and 5)
= 31 × 51
Factors with their highest powers are 22, 31, 51
Taking products of factors with their highest powers = 22 × 31 × 51
∴ LCM of 12 and 15 = 22 × 31 × 51 = 60

## History

Fundamental Theorem of Arithmetic is also known by other names as Unique Factorization Theorem and Prime Factorization Theorem. The theorem can be derived from the statements made by the mathematician Euclid in his book VIII of Elements in proposition 30, proposition 31 and proposition 32.

It was Kamal al-Din Hasan ibn Ali ibn Hasan al-Farisi who introduced fundamental theorem of arithmetic first time from Euclid’s propositions.

Carl Friedrich Gauss took the next steps to give the proof of fundamental theorem of arithmetic in his book Disquisitiones Arithmeticae.

Last updated on: 27-07-2024