

How to State and Prove the Fundamental Theorem of Arithmetic?
The concept of Fundamental Theorem of Arithmetic plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It is especially crucial for factorization, HCF, and LCM operations—core topics for board exams, Olympiads, and higher studies.
What Is Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers, except for the order of the primes. This means that the factorization of any number into prime numbers is unique. You'll find this concept applied in areas such as Prime Numbers study, calculations, and the Number System chapter.
Key Formula for Fundamental Theorem of Arithmetic
Here’s the standard formula: For any integer \( n > 1 \),
\( n = p_1^{a_1} \times p_2^{a_2} \times \dotsm \times p_k^{a_k} \),
where each \( p_i \) is a prime number and \( a_i \) is a positive integer.
This representation is unique except for the order of the primes.
Cross-Disciplinary Usage
Fundamental Theorem of Arithmetic is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE, NEET, and competitive exams will see its relevance in number theory, cryptography, data compression, and coding theory.
Step-by-Step Illustration
- Take the number 140.
Divide by the smallest prime (2): 140 ÷ 2 = 70 - Divide the result again by 2: 70 ÷ 2 = 35
- Next, try 3 (does not divide), then 5: 35 ÷ 5 = 7
- 7 is a prime, so the process stops.
So, 140 = 2 × 2 × 5 × 7 - In any order, the prime factors are always the same. This shows uniqueness!
Number | Prime Factors |
---|---|
156 | 2 × 2 × 3 × 13 |
234 | 2 × 3 × 3 × 13 |
91 | 7 × 13 |
Speed Trick or Vedic Shortcut
Here’s a quick method to check if a number is prime or to factorize fast: Start dividing by 2, then 3, 5, 7…and so on, until you reach a prime greater than the square root of the number. Many use "factor trees" to finish factorization in seconds—great for exams!
Example Trick: To quickly factorize 210:
Start with 2: 210 ÷ 2 = 105 → 210 = 2 × 105
Next try 3 for 105: 105 ÷ 3 = 35 → 210 = 2 × 3 × 35
Next, 5 for 35: 35 ÷ 5 = 7 → 210 = 2 × 3 × 5 × 7
All are primes! Done.
Such tricks help you save time during board exams and Olympiads. Vedantu teachers share several more shortcuts in their live classes to help you master Maths questions quickly.
Try These Yourself
- Express 420 as a product of prime factors using the factor tree technique.
- Find the unique prime factorization of 252.
- Is the number 73 a prime, or composite? Justify using the fundamental theorem of arithmetic.
- Find HCF and LCM of 18 and 24 using prime factorization.
Frequent Errors and Misunderstandings
- Remember, the order of prime factors doesn't matter—2 × 3 × 5 is the same as 5 × 3 × 2.
- Some students confuse factors with prime factors—always factorize down to primes.
- Don’t include 1 as a prime factor. 1 is neither prime nor composite.
Relation to Other Concepts
The idea of the fundamental theorem of arithmetic is closely linked with prime factorization, composite numbers, factorisation, and Euclid’s division lemma. Understanding this theorem helps in mastering more advanced concepts like LCM, HCF and even cryptography basics, as used in modern computer science.
Classroom Tip
A helpful way to remember the Fundamental Theorem of Arithmetic is by picturing numbers as unique towers built out of prime “building blocks.” If you draw a factor tree for any number, you’ll always get the same prime “roots” no matter which branches you choose—just in a different order! Vedantu teachers use diagrams and stories like this to make Maths topics lively and memorable.
Wrapping It All Up
We explored the Fundamental Theorem of Arithmetic—from its definition, formula, step-by-step examples, exam tricks, common mistakes, and related concepts. Keep practicing such problems to boost your confidence in Maths. For more clear explanations, solved problems, and revision notes, check out Vedantu’s online classes—you’ll soon be a pro at recognizing unique prime factorization in any exam or real-life problem!
Further Learning — Useful Internal Links
FAQs on Fundamental Theorem of Arithmetic Explained with Proof & Applications
1. What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, disregarding the order of the factors. This means each composite number has one and only one prime factorization.
2. How do you state and prove the Fundamental Theorem of Arithmetic?
Statement: Every integer greater than 1 is either a prime number itself or can be uniquely expressed as a product of prime numbers.
Proof: The proof typically involves the Euclidean algorithm and the concept of unique factorization. A detailed, step-by-step proof is available on the Vedantu page.
3. What are the applications of the Fundamental Theorem of Arithmetic?
The theorem is crucial for finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers. It also has applications in areas like cryptography and abstract algebra.
4. Can prime factorization be unique for every number?
Yes, the Fundamental Theorem of Arithmetic guarantees the uniqueness of prime factorization for every integer greater than 1. The order of the prime factors might differ, but the factors themselves will always be the same.
5. How is the Fundamental Theorem of Arithmetic used to find the HCF and LCM of two numbers?
To find the HCF, identify the common prime factors with the lowest powers. To find the LCM, identify all prime factors with their highest powers. Multiply these to obtain the LCM.
6. What is a prime factorization, and how do I perform it?
Prime factorization is the process of expressing a number as a product of its prime factors. Methods include the factor tree and trial division. Examples are provided on the Vedantu page.
7. Is the Fundamental Theorem of Arithmetic part of the standard math curriculum?
Yes, it's a fundamental concept in number theory, usually covered in secondary school mathematics.
8. What is the difference between prime and composite numbers?
A prime number is a natural number greater than 1 that has only two divisors: 1 and itself. A composite number is a natural number greater than 1 that is not prime (it has more than two divisors).
9. Are there any exceptions to the Fundamental Theorem of Arithmetic?
No, the theorem holds true for all integers greater than 1. There are no exceptions.
10. How does the Fundamental Theorem of Arithmetic relate to the Unique Factorization Theorem?
They are essentially the same theorem; the Fundamental Theorem of Arithmetic is just a more commonly used name for the Unique Factorization Theorem in number theory.
11. Why is unique prime factorization important in mathematics?
Unique prime factorization is essential for many mathematical concepts and proofs, providing a foundation for understanding number properties and solving various problems.
12. Can you give an example of how to use the Fundamental Theorem of Arithmetic to solve a problem?
Find the HCF and LCM of 12 and 18. First, find the prime factorizations: 12 = 2² x 3; 18 = 2 x 3². The HCF is 2 x 3 = 6. The LCM is 2² x 3² = 36.

















