Mathematical Algorithms | Prime Factorization and Divisors

Divisors:

Mathematically, divisor is defined as a number that divides another number completely or with a remainder. 

For example: Consider 6 / 3. Here 3 divides 6 completely. Also, consider 8 /3. Here 3 does not divide 8 completely. But in both cases, 3 is the divisor.

Prime Factorization:

Prime factorization is a very common topic of mathematics and is frequently used in several algorithms to solve problems. Prime factorization of a number is the process of finding which prime numbers multiply together to form the original number.

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.

Some practice problems on Prime Factorization and Divisors:

• Easy:
  1. Prime factors
  2. Fermat’s Last Theorem
  3. Sphenic Number
  4. Hoax Number
  5. Blum Integer
  6. Superperfect Number
  7. Deficient Number
  8. k-Rough Number or k-Jagged Number
  9. Perfect Number
  10. Aliquot sum
  11. Find all factors of a Natural Number
  12. Sum of all the factors of a number
  13. Total number of divisors for a given number
  14. Check if count of divisors is even or odd
• Intermediate:
  1. Frugal Number
  2. Betrothed numbers
  3. Lemoine’s Conjecture
  4. k-th prime factor of a given number
  5. Prime Factorization using Sieve O(log n) for multiple queries
  6. Finding power of prime number p in n!
  7. Program for Mobius Function
  8. Ordered Prime Signature
  9. Fast inverse square root
  10. Maximum number of unique prime factors
  11. Common Divisors of Two Numbers
  12. Find largest prime factor of a number
  13. Find if n can be written as product of k numbers
  14. Product of unique prime factors of a number
  15. Check whether a number has exactly three distinct factors or not
  16. Determine whether a given number is a Hyperperfect Number
• Hard:
  1. Smith Numbers
  2. Find numbers with n-divisors in a given range
  3. Pollard’s Rho Algorithm for Prime Factorization
  4. P-Smooth Numbers or P-friable Number
  5. Find sum of divisors of all the divisors of a natural number
  6. No of Factors of n!
  7. Print all prime factors and their powers
  8. Generation of n numbers with given set of factors
  9. Efficient program to print the number of factors of n numbers
  10. Ways to express a number as product of two different factors
  11. Sum of all divisors from 1 to n
  12. Prime factors of a big number
  13. Count Divisors of n in O(n^1/3)

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