Theorem 1.1 - Fundamental Theorem of Arithmetic
Statement: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Given: Let N be a composite number. It can be expressed as a product of prime numbers.
To Prove: The prime factorization of N is unique, apart from the order of factors
Proof:
Conclusion: The Fundamental Theorem of Arithmetic ensures that every composite number has a unique prime factorization, apart from the order of the factors
Statement: If a prime number p divides a², then p also divides a, where a is a positive integer.
Given: Let p be a prime number and a be a positive integer such that p divides a².
To Prove: If p divides a², then p must also divide a.
Proof:
The Theorem 1.2 proves that if a prime number divides the square of a number, then it must also divide the number itself. This theorem is widely used in number theory and proofs of irrationality, such as proving that √2 is irrational.
Statement: √2 is an irrational number.
To Prove: √2 cannot be expressed as a fraction pq, where p and q are integers and q ≠ 0.
Proof :
Conclusion: Thus, √2 is irrational.
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1. What is the Fundamental Theorem of Arithmetic? | ![]() |
2. How do Theorems 1.2 and 1.3 relate to the Fundamental Theorem of Arithmetic? | ![]() |
3. Why is the uniqueness of prime factorization important in mathematics? | ![]() |
4. Can the Fundamental Theorem of Arithmetic be applied to negative integers or zero? | ![]() |
5. How is the Fundamental Theorem of Arithmetic used in real-world applications? | ![]() |