Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is a key result in number theory that explains how every whole number (greater than 1) can be built using prime numbers.

It is also known as the unique factorization theorem and prime factorization theorem

The figures above represents the factorization of different numbers, which shows that a composite numbers can be expressed as a product of prime numbers. If we keep on trying different numbers, we see that all the numbers can be represented as product primes.

Proof of Fundamental Theorem of Arithmetic

Theorem: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Proof:

Step 1: The existence of prime factors, we will prove it b y induction

Firstly, consider n > 1

Therefore Initially, n = 2. Since n = 2 and 2 is a prime number, the result is true.

Consider n > 2  (Induction hypothesis: Let result be true for all positive numbers less than n )

No, we will prove that the result is also true for n. 

• If n is prime, then n is a product of primes is trivially true.
• If n is not prime i.e n is a composite number, then 

n = ab, a, b < n

By induction method, the result is true for a and b (because a<n and b<n). Therefore, by the induction hypothesis, a must be the product of prime numbers and b is a product of prime numbers. Therefore, n = ab is a product of prime numbers. Thus, it is proved by induction.

Step 2: Uniqueness (of factors up to order)

Let n = p₁ p₂ p₃....p_k (where p₁ p₂ p₃ ...p_k are primes)

Let if possible, there be two representations of n as a product of primes

i.e let if possible n=p₁ p₂ p₃....p_k = q₁ q₂ q₃ ...q_r      where p_i's and q_j's are prime numbers 

(we will prove that pi's are the same as qj's )

Now p₁/p₁ p₂ p₃....p_k, Therefore, p1/q₁ q₂ q₃ ...q_r) (because p₁ p₂ p₃....p_k = q₁ q₂ q₃ ...q_r)

Therefore, by result p1 must be one of the qj's.

Let p₁ = q₁

So we get p₁ p₂ p₃....p_k = q₁ q₂ q₃ ...q_r

= q₁ q₂ q₃...q_r 

And by cancellation p1 from both sides, 

p₂ p₃....p_k  = q₁ q₂ q₃ ...q_r

So by the same argument, we will get p₂=q₂  and so on.

Thus, n can be expressed as a product of primes uniquely (except for the order)

Hence proved.

Related Articles

• Real Life Applications of Prime Factorization
• Euclid Theorem
• Theorems Related to Prime Numbers

Examples On Fundamental Theorem of Arithmetic

Question 1: Factorize the number “4072” and represent it as product of primes. 

Answer: 

Given Number = 4072

Prime factorization of 4072 = 2 × 2 × 2 × 509

Question 2: Factorize the number “324” and represent it as product of primes. 

Answer:

Given Number = 324

Prime factorization of 324= 2 × 2 × 3 × 3 × 3 × 3

Question 3: Factorize the number “16048” and represent it as product of primes. 

Answer: 

Given Number = 16048

Prime factorization of 16048= 2 × 2 × 2 × 2 × 17 × 59

LCM and HCF using Fundamental Theorem of Arithmetic

• HCF known as the highest common factor is the greatest number that divides each of the two numbers given.
• LCM is the lowest common multiple that is the product of all the common prime factors but with their highest degrees/powers.

Example Questions on LCM and HCF using Fundamental Theorem of Arithmetic

Question 1: Find the LCM and HCF of 24 and 36.

The Prime factors of 24 = 2 × 2 × 2 × 3

The prime factors of 36 = 2 × 2 × 3 × 3

HCF = 2 × 2 × 2 × 3, 2 × 2 × 3 × 3 = 2 × 2 × 3 = 12

LCM =  2 × 2 × 2 × 3 × 3 = 72

LCM and HCF can be found with the help of prime factorization too, let's look at some examples.

Question 2: Find the LCM and HCF of numbers 6 and 20. 

Prime Factorization of 6 can be represented in the following way, 

Prime Factorization of 20 can be represented in the following way, 

So, now we have prime factorization of both the numbers, 

6 = 2 × 3

20 = 2 × 2 × 5 

We know that 

HCF = Product of the smallest power of each common prime factor in the numbers.

LCM = Product of the greatest power of each prime factor, involved in the numbers.

So, HCF(6,20) = 2¹

      LCM(6,20) = 2² × 3¹ × 5

Question 3: Find the LCM and HCF of numbers 24 and 36. 

Prime Factorization of 24: 

Prime Factorization of 36: 

24 = 2^{3 ×} 3 and 36 = 2² × 3²

Based on the previous definitions,  

HCF(24, 36) = 12 

LCM(24, 36) = 72

Fact: In the above examples, notice that for any two numbers “a” and “b”. HCF × LCM = a × b. 

Practice Questions on Fundamental Theorem of Arithmetic

Question 1. Perform Prime factorization to express 630 as product of primes.

Question 2. Find the least common multiple of 36 and 48 using the prime factorization method.

Question 3. Find the greatest common divisor of 84 and 126 using the prime factorization method.

Question 4. Prove that if a number ends in 0, then it is divisible by 10 using the Fundamental Theorem of Arithmetic.

Question 5. Prove that every positive integer greater than 1 can be written uniquely as a product of prime numbers, up to the order of the factors.