2. Let n be a positive integer, and let A be a list of positive integers. We say that the integer n can be factorized by A if there exists a sequence of integers 11, 12,..., ik (with 0 ≤i; < len (A), for j = 1,..., k) such that n is the product of the integers A[i], A[i2],..., A[ik]. Write an efficient algorithm factorizable (n, A) that returns True if n can be factorized by A, and False otherwise. Prove that your algorithm is correct, and bound its running time. Larger scores will be awarded to faster solutions. Example 1: if n = 6 and A [4,2,3], then factorizable (n, A) should return True, L[1] * L[2]. since n == Example 2: if n = 8 and A = L[0] * L[0] * L[0]. [2,5], then factorizable (n, A) should return True, since ,2,3,5,7], then factorizable (n, A) should return Example 3: if n = 13 and A = False since n cannot be factorized by A. **Algorithm and Factorization in Positive Integers****Problem Statement:**Consider a positive integer \( n \) and a list \( A \) of positive integers. The integer \( n \) can be factorized by the elements in \( A \) if there exists a sequence of integers \( i_1, i_2, \ldots, i_k \) (with \( 0 \le j < \text{len}(A) \) for \( j = 1, \ldots, k \)) such that \( n \) is the product of the integers \( A[i_1], A[i_2], \ldots, A[i_k] \).**Task:**Develop an efficient algorithm `factorizable(n, A)` that returns `True` if \( n \) can be factorized by \( A \), and `False` otherwise. Additionally, prove the correctness of your algorithm and provide an analysis of its runtime. Note that faster solutions will be awarded higher scores.**Examples:**1. **Example 1:** If \( n = 6 \) and \( A = [4, 2, 3] \), - Here, \( n \) can be expressed as the product of the factors 2 and 3, both of which exist in the list \( A \). - Hence, `factorizable(n, A)` should return `True`.2. **Example 2:** If \( n = 8 \) and \( A = [2, 5] \), - Here, \( n \) can be expressed as the product of the factors 2, 2, and 2 (i.e., \( 2^3 \)), all of which are present in the list \( A \). - Hence, `factorizable(n, A)` should return `True`.3. **Example 3:** If \( n = 13 \) and \( A = [1, 2, 3, 5, 7]`, - Here, \( n = 13 \) cannot be expressed as the product of any combination of factors in the list \( A \). - Hence, `factorizable(n, A)` should return `False`.

The algorithm works by checking if n can be factorized using the numbers in A. It starts with the…

Answered: 2. Let n be a positive integer, and let…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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2. Let n be a positive integer, and let A be a list of positive integers. We say that the integer n can be factorized by A if there exists a sequence of integers 11, 12,..., ik (with 0 ≤i; < len (A), for j = 1,..., k) such that n is the product of the integers A[i], A[i2],..., A[ik]. Write an efficient algorithm factorizable (n, A) that returns True if n can be factorized by A, and False otherwise. Prove that your algorithm is correct, and bound its running time. Larger scores will be awarded to faster solutions. Example 1: if n = 6 and A [4,2,3], then factorizable (n, A) should return True, L[1] * L[2]. since n == Example 2: if n = 8 and A = L[0] * L[0] * L[0]. [2,5], then factorizable (n, A) should return True, since ,2,3,5,7], then factorizable (n, A) should return Example 3: if n = 13 and A = False since n cannot be factorized by A.

**Algorithm and Factorization in Positive Integers**

**Problem Statement:**

Consider a positive integer \( n \) and a list \( A \) of positive integers. The integer \( n \) can be factorized by the elements in \( A \) if there exists a sequence of integers \( i_1, i_2, \ldots, i_k \) (with \( 0 \le j < \text{len}(A) \) for \( j = 1, \ldots, k \)) such that \( n \) is the product of the integers \( A[i_1], A[i_2], \ldots, A[i_k] \).

**Task:**

Develop an efficient algorithm `factorizable(n, A)` that returns `True` if \( n \) can be factorized by \( A \), and `False` otherwise. Additionally, prove the correctness of your algorithm and provide an analysis of its runtime. Note that faster solutions will be awarded higher scores.

**Examples:**

1. **Example 1:** If \( n = 6 \) and \( A = [4, 2, 3] \),

- Here, \( n \) can be expressed as the product of the factors 2 and 3, both of which exist in the list \( A \).
- Hence, `factorizable(n, A)` should return `True`.

2. **Example 2:** If \( n = 8 \) and \( A = [2, 5] \),

- Here, \( n \) can be expressed as the product of the factors 2, 2, and 2 (i.e., \( 2^3 \)), all of which are present in the list \( A \).
- Hence, `factorizable(n, A)` should return `True`.

3. **Example 3:** If \( n = 13 \) and \( A = [1, 2, 3, 5, 7]`,

- Here, \( n = 13 \) cannot be expressed as the product of any combination of factors in the list \( A \).
- Hence, `factorizable(n, A)` should return `False`.

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