This is an example of an inline equation: \(E = mc^2\)
This is an example of a block equation:
\[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \] - Let ( n ) be any positive integer, and let \( p_1, p_2, \ldots, p_k )\ be all the distinct prime numbers that divide ( n ). Then there exists a unique set of non-negative integers ( \alpha_i ) such that: \[ n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}. \] 2. Proof: - Existence: We can prove this by using the Fundamental Theorem of Arithmetic (also known as the unique prime factorization theorem). According to the Fundamental Theorem of Arithmetic, every integer ( n ) greater than 1 has a unique prime factorization. Thus, for any positive integer ( n ), we can write: \[ n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}, \] where each ( p_i ) is a prime number and each ( \alpha_i \geq 0 ). This shows that such a factorization exists. - Uniqueness: Suppose, for contradiction, that there exist two different factorizations of ( n ): \[ n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}, \] and \[ n = q_1^{\beta_1} q_2^{\beta_2} \cdots q_r^{\beta_r}, \] where ( p_i ) and ( q_j ) are prime numbers, and ( \alpha_i, \beta_j \geq 0 ). Because of the uniqueness of the prime factorization of ( n ), each prime ( p_i ) must be equal to some ( q_j ), and similarly, each ( q_j ) must be equal to some ( p_i ). Consequently, we can rearrange the factors so that: \[ {p_1, p_2, \ldots, p_k} = {q_1, q_2, \ldots, q_r}. \] Additionally, for each ( i ), ( p_i ) must appear the same number of times in both factorizations, implying that: \[ \alpha_i = \beta_j. \] Therefore, the two factorizations are identical, contradicting our assumption of their difference. This proves the uniqueness. 3. Conclusion: - We have established both the existence and uniqueness of the prime factorization for any positive integer ( n ). Thus, we conclude that every positive integer can be uniquely expressed as a product of prime powers. \[ \boxed{\text{Theorem: Every positive integer } n \text{ has a unique prime factorization.}} \]