Fundamental Theorem of Arithmetic

According to the fundamental theorem of arithmetic, the Factorization of each and every composite number can be expressed as a product of prime numbers regardless of the sequence wherein the prime major determinants of that particular number appear. The fundamental theorem of arithmetic is an extremely useful tool for understanding any number’s greatest common divisor.

Fundamental Theorem of Arithmetic Definition

Every composite integer could be factorized as just a product of prime numbers, and so this factorization is distinct, apart from the sequence wherein the major determinants of prime numbers appear is as per the fundamental theorem of arithmetic.

Let’s look at the prime factorization of 210, for example.

Factors of 210 = 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

Prime Factorization of 210 = 2 × 3 × 5 × 7.

From the above example, we get 210 = 2 × 3 × 5×7. This theorem also states that such a factorization must be one-of-a-kind. In other words, there really is no alternative method to describe 210 as a prime product. Of course, the order in which the prime factors appear can be altered. The prime factorization, for example, can be represented as:

 210 = 21 × 31 × 51 × 71. However, the set of prime factors (as well as the frequency with which each component appears) is unique. That is, there can only be one prime factorization for 210 with one factor of 2 that is 21, one factor of 3 that is 31, one factor of 5 that is 51, and one factor of 7 that is 71.

Example 2: Let’s look at the prime factorization of 350 = 21 × 52 × 71 where 2, 5, 7 are considered prime numbers.

 Example 3: Let’s look at the prime factorization of 1200 = 24 × 31 × 52 where 2, 3, 5 are considered prime numbers.

 Example 4: Let’s look at the prime factorization of 140 = 22 × 5 × 7 where 2, 5, 7 are considered prime numbers.

 Example 5: Let’s look at the prime factorization of 24 = 2 × 2 × 2 ×3  where 2, 3 are considered prime numbers.

Proof for Fundamental Theorem of Arithmetic

We must explain the presence and uniqueness of the prime numbers in an attempt to validate the foundational theorem of arithmetic. As a result, the demonstration of the fundamental theorem of arithmetic is conducted in two parts. We’ll first consider that any integer, n ≥2, may be written in a different manner as the product of primes: n =  p1x  p2x …….x   pii.

Step – Explaining the existence of Prime Factorization

We will prove this instance with mathematical induction.

Basic Step: Let’s consider the statement is correct for n = 2.

Steps of assumption: Let us derive assumptions that the statement is correct for n = k.

The product of primes can thus be written as k.

Step 1: Let’s show that the statement is correct for n = k + 1.

The case is evident if k + 1 is a prime number.

If k + 1 is not a prime number, it must have a prime factor, such as p.

Then k + 1 =  pj, where j< k (1)

Since j< k, k can be represented as the product of primes using the “inductive step.”

As a result of (1), k + 1 can also be expressed as a prime product. The “presence of factorization” is thus proven through mathematical induction.

A composite number is written as a product of prime numbers in mathematical logic, and so this factorization is distinct excluding the sequence wherein the prime factors exist.

We could further see in this theorem that not only would a composite number be factored as the product of its prime numbers, but also that the factorization is distinct for every composite number, irrespective of the order wherein the major determinants of prime factors appear.

To put it differently, there will be only one way of representing a natural number using prime factors. This concept can alternatively be expressed as follows:

Except for the sequence of their factors, every natural number’s prime factorization is considered to be unique.

Fundamental Theorem of Arithmetic Examples

Question: In a  sporting event, two competing cars A and B take 30 minutes and 45 minutes to accomplish one round of the racetrack, respectively. How long will it take for the cars to reassemble at the starting point?

Answer: Because car B takes longer to finish one round than car A, it’s reasonable to infer that A will reach first, and the two vehicles would encounter again after A has returned to the starting point. Determining the L.C.M of the time it takes for each could be used to calculate this duration.

30 = 2 × 3 × 5 (Prime Numbers)

45 = 3 × 3 × 5 (Prime Numbers)

The L.C.M is 90 in the above case

As a result, after 90 minutes, both cars will reunite at the starting position.