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174 LearnersLast updated on September 10, 2025
Fundamental Theorem of Arithmetic
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.
Proof of the Fundamental Theorem of Arithmetic
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:
- If n is a prime, its factorization is n itself. So, it is unique.
- If n is a non-prime, it can be expressed as a product of two smaller integers:
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.
Using the Fundamental Theorem of Arithmetic for HCF and LCM
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.
Common mistakes and How to Avoid Them in Fundamental Theorem of Arithmetic
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:
Mistake 1
Confusing Prime and Composite Numbers
Students must always break a number down until the prime numbers remain. They must also remember that a prime number is only divisible by 1 and itself. They can verify each factor by checking whether it has only two divisors.
Mistake 2
Not Checking With the Smallest Prime Number
It is important to start with 2, the smallest prime number, and then continue with other primes.
Mistake 3
Missing Out Prime Factors.
We should count each and every prime factor carefully and use proper exponent notation to express the recurring factors. We can also cross-verify the answer by multiplying the factors and checking if it matches the original number.
Mistake 4
Not Considering the Uniqueness of Prime Factorization
We should always remember that prime factorization is unique except for the order. If this rule is not satisfied, then the theorem cannot be proved.
Mistake 5
Stopping Factorization Too Early
Students must keep breaking down the composite factors until only primes remain. They can use the factor tree or division method systematically.
Real life applications of 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:
- Cryptography and Cybersecurity:
Modern encryption systems rely on the fundamental theorem of arithmetic. Since the prime factorization is unique and non-trivial for large numbers, this property is used to create secure encryption keys that protect sensitive data in banking transactions, online communication, and digital signatures.
- Error Detection and Correction in Data Transmission:
Digital communication systems, such as mobile networks, Wi-Fi signals, and barcode scanners, use error detection and correction codes to ensure that the data that has to be transmitted is transmitted securely and accurately. Many of these systems rely on prime factorization to verify data integrity.
- Computer Science and Algorithm Optimization:
Prime factorization plays a vital role in optimizing algorithms, particularly those based on number theory. In computer science, it is used in hashing functions and pseudorandom number generators. For example, hashing functions and pseudorandom number generators rely on prime numbers to ensure unpredictability and uniqueness, which are essential for secure computing and data retrieval.
Solved examples on Fundamental Theorem of Arithmetic
Problem 1
Find the prime factorization of 30.
The prime factorization of 30 is 2 x 3 x 5.
Explanation
Divide by 2,
30 ÷ 2 = 15
Divide by 3,
15 ÷ 3 = 5
Conclude with a prime number:
5 is a prime number.
Problem 2
Determine the prime factorization of 60.
The prime factorization of 60 = 2² × 3 × 5.
Explanation
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.
Problem 3
Find the prime factors of 84
The prime factors of 84 = 2² × 3 × 7.
Explanation
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.
Problem 4
Determine the prime factorization of 90.
The prime factorization of 90 = 2 × 3² × 5.
Explanation
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.
Problem 5
Find the prime factorization of 105.
The prime factorization of 105 is 3 x 5 x 7
Explanation
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.
FAQs on Fundamental Theorem of Arithmetic
1.What is the fundamental theorem of arithmetic?
2.Why is it called “fundamental”?
3.What exactly are prime numbers?
4.Does the theorem apply to the number 1?
5.What does “uniqueness” mean in this context?
6.How can families in United States create number-rich environments to improve Fundamental Theorem of Arithmetic skills?
7.What are some fun ways kids in United States can practice Fundamental Theorem of Arithmetic with numbers?
8.What role do numbers and Fundamental Theorem of Arithmetic play in helping children in United States develop problem-solving skills?
9.How can children in United States use numbers in everyday life to understand Fundamental Theorem of Arithmetic?
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Hiralee Lalitkumar Makwana
About the Author
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.
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: She loves to read number jokes and games.
