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Last updated on October 9, 2025
According to the Fundamental Theorem of Arithmetic, every whole number greater than 1 can be uniquely expressed as a product of prime numbers. The theorem also states that the order of the prime factors does not affect the outcome.
The proof to find the fundamental theorem of arithmetic is given below:
Theorem: Every integer greater than 1 can be written as a product of prime numbers. Although the factorization is unique, the order of the prime factors can be different.
Proof:
Step 1: Use mathematical induction to prove that every whole number greater than 1 has at least one prime factor.
First, consider any integer n where n > 1
Now consider n = 2. Since 2 is a prime number and greater than 1, n > 1 holds true.
Let us assume that all whole numbers less than n have at least one prime factor.
Now we shall prove that the statement is also true for n.
n is already a product of prime numbers if n is a prime.
n is a composite number if it is not a prime. This means that it can be expressed as a product of smaller numbers.
Step 2: Proving Uniqueness
In the second step, we will prove that the factorization is unique. Once again, we will use mathematical induction.
Let n = 2. Here, the only prime factorization is 2 itself, and hence it is unique.
Now, let us assume that for all whole numbers below n, their prime factorization is unique.
To prove that the uniqueness holds for n, let us consider the following statements:
n = a × b
Since a and b are lesser in value than n, both a and b have a unique prime factorization as per the inductive hypothesis.
Therefore, the product of a and b (which equals n) also has a unique prime factorization. Also, without rearranging the same factors, n cannot be formed.
Thus, the fundamental theorem of arithmetic is proved.
We use fundamental theorem of arithmetic to find the HCF and LCM of two or more numbers.
Let us see how to find them:
HCF can be determined by finding the product of the smallest power of each common prime factor. If we can find the product of the greatest power of each prime factor, then LCM can be determined.
Let us understand this by an example:
Find the HCF of 120 and 180.
First, find the prime factorization of 120:
Prime factorization of 120 = 2³ × 3¹ × 5¹
Prime factorization of 180 = 2², 3¹, 5¹
To find the HCF, we find the product of the smallest power of each common prime factor:
Common factors = 2, 3, 5
Smallest powers = 2², 3¹, 5¹.
Hence, HCF = 22 x 31 x 51
= 4 x 3 x 5
= 60
Since the LCM is found by multiplying the greatest powers of each prime factor:
LCM = 23 x 32 x 51
= 8 x 9 x 5
= 360.
Students tend to make mistakes while understanding the concept of the fundamental theorem of arithmetic. Let us see some common mistakes and how to avoid them, in fundamental theorem of arithmetic:
The fundamental theorem of arithmetic has numerous applications across various fields. Let us explore how the fundamental theorem of arithmetic is used in different areas:
Find the prime factorization of 30.
The prime factorization of 30 is 2 x 3 x 5.
Divide by 2,
30 ÷ 2 = 15
Divide by 3,
15 ÷ 3 = 5
Conclude with a prime number:
5 is a prime number.
Determine the prime factorization of 60.
The prime factorization of 60 = 2² × 3 × 5.
Divide by 2,
60÷2 = 30
Divide by 2 again:
30÷2 = 15.
Divide by 3,
15÷3 = 5
Conclude with a prime:
5 is a prime number.
Find the prime factors of 84
The prime factors of 84 = 2² × 3 × 7.
Divide by 2,
84 ÷ 2 = 42
Divide by 2 again:
42 ÷ 2 = 21
Divide by 3,
21 ÷ 3 = 7
Conclude with a prime:
7 is a prime number.
Determine the prime factorization of 90.
The prime factorization of 90 = 2 × 3² × 5.
Divide by 2,
90 ÷ 2 = 45
Divide by 3,
45 ÷ 3 = 15
Divide by 3 again:
15 ÷ 3 = 5
Conclude with a prime:
5 is a prime number.
Find the prime factorization of 105.
The prime factorization of 105 is 3 x 5 x 7
Test divisibility by 2,
105 is odd, so skip 2.
Divide by 3,
105 ÷ 3 = 35
Divide by 5,
35 ÷ 5 = 7
Conclude with a prime:
7 is a prime number.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.