Mathematical Algorithms - Prime Factorization and Divisors

In the Mathematical algorithm there are two fundamental mathematical algorithms that is prime factorization and finding divisors. Prime factorization is the process of breaking down a number into its prime factors, while finding divisors involves identifying all the numbers that can evenly divide the original number. These concepts are very important in various mathematical and computer science applications, from number theory to cryptography. In this article we will look into these algorithms with some practice problem.

Divisors:

Divisors are the numbers that divide evenly into a given number without leaving a remainder.

For example, the divisors of 12 are: 1, 2, 3, 4, 6, and 12. These are all the numbers that can divide 12 with no remainder left over.

Finding the divisors of a number is an important mathematical operation. It has applications in number theory, computer science algorithms, and more.

Some key things to know about divisors:

• Every number is a divisor of itself and 1.
• Smaller numbers tend to have more divisors than larger numbers.
• Prime numbers only have two divisors - 1 and the number itself.
• Efficiently finding all the divisors of a number is an important algorithmic problem.

Prime Factorization:

Prime factorization is the process of breaking down a number into its prime factors. A prime number is a positive integer greater than 1 that is only divisible by 1 and itself. The prime factorization of a number involves finding which prime numbers, when multiplied together, equal that original number.

For example, the prime factorization of 24 is 2 x 2 x 2 x 3. This is because the prime numbers that, when multiplied together, equal 24 are 2, 2, 2, and 3.

Some key points about prime factorization:

• Every positive integer greater than 1 has a unique prime factorization.
• Prime factorization is useful for finding the divisors of a number.
• Efficient algorithms for prime factorization are important in areas like cryptography.
• Prime factorization can reveal insights about the structure and properties of numbers.

Mathematically, any number N can be expressed as:

N = p₁^e₁ * p₂^e₂ * p₃^e₃ * . . . where p₁, p₂, p₃ are prime numbers and e₁, e₂, e₃ are the exponents of the prime numbers.

Finding these primes for any number N is called prime factorization.

Easy Problems on Prime Factorization and Divisors:

• Prime factors
• Fermat’s Last Theorem
• Sphenic Number
• Hoax Number
• Blum Integer
• Superperfect Number
• Deficient Number
• k-Rough Number or k-Jagged Number
• Perfect Number
• Aliquot sum
• Find all factors of a Natural Number
• Sum of all the factors of a number
• Total number of divisors for a given number
• Check if count of divisors is even or odd

Medium Problems on Prime Factorization and Divisors:

• Frugal Number
• Betrothed numbers
• Lemoine’s Conjecture
• k-th prime factor of a given number
• Prime Factorization using Sieve O(log n) for multiple queries
• Finding power of prime number p in n!
• Program for Mobius Function
• Ordered Prime Signature
• Fast inverse square root
• Maximum number of unique prime factors
• Common Divisors of Two Numbers
• Find largest prime factor of a number
• Find if n can be written as product of k numbers
• Product of unique prime factors of a number
• Check whether a number has exactly three distinct factors or not
• Determine whether a given number is a Hyperperfect Number

Hard Problems on Prime Factorization and Divisors:

• Smith Numbers
• Find numbers with n-divisors in a given range
• Pollard’s Rho Algorithm for Prime Factorization
• P-Smooth Numbers or P-friable Number
• Find sum of divisors of all the divisors of a natural number
• No of Factors of n!
• Print all prime factors and their powers
• Generation of n numbers with given set of factors
• Efficient program to print the number of factors of n numbers
• Ways to express a number as product of two different factors
• Sum of all divisors from 1 to n
• Prime factors of a big number
• Count Divisors of n in O(n^1/3)