Given an integer N. Find the minimum number of square free divisors. In other words, the factorization of N should comprise of only those divisors that are square free. A square free number is that number which is not divisible by any perfect square (Of course, 1 will not be considered as perfect square in this case).
Constraints: 2 ≤ N ≤ 106
Input : 24
Output : 3
Explanation: 24 will be represented as 24 = 6 x 2 x 2, here every factor
(6 and 2) both are square free.
Note: 24 cannot be represented as (24 = 12 x 2) or (24 = 24) or (24 = 8 x 3) because in all these factorizations, every factorization has atleast one number that is not square free i.e. 12, 24 and 8 are divisible by 4 (4 is a perfect square). Hence minimum number of square free divisors is 3.
Input : 6
Output : 1
Explanation: 6 will be represented as 6 = 6, since 6 is not divisible by any perfect square. Hence minimum number of square free divisors is 1.
Prerequisites: Sieve of Eratosthenes
A Naive Approach is to consider all possible factorizations of number N and then check if there exists a number which is not square free (divisible by some perfect square) for each factorization. If it exists, then discard that factorization, otherwise consider that factorization in answer. After considering all correct factorizations, find the minimum number of square free divisors.
Efficient Approach: Build Prime Sieve (using Sieve of Eratosthenes) by finding all the prime factors upto square root of N (upto 103 considering the constraints). Now, consider all prime factors less than or equal to square root of N and for each prime factor find its maximum power in number N (like max power of 2 in 24 is 3). Now, we know that if a prime factor has a power greater than 1 in N, it can’t be grouped with itself (for e.g. 2 has power of 3 in 24, hence 2 x 2 = 4 or 2 x 2 x 2 = 8 can’t occur in the factorization of 24 as both of them are not square free) since it will be divisible by some perfect square. But a prime factor grouped with another prime factor (only once) will never be divisible by any perfect square. This gives us an intuition that answer will be the maximum of maximum powers of all prime factors in number N. In other words,
If prime factorization of N = f1p1 * f2p2 * ... * fnpn where f1, f2 ... fn are the prime factors and p1, p2 ... pn are their maximum respective powers in N. Then answer will be, ans = max(p1, p2, ..., pn) For e.g. 24 = 23 * 31 ans = max(3, 1) = 3
Below is an illustration to explain the above algorithm
Let N = 24. Factorization of N can be represented in many ways and out of those we have to find that in which there are minimum divisors with each divisor being square free. Some factorizations are:
24 = 24 (24 is not square Free)
24 = 2 * 12 (12 is not square free)
24 = 3 * 8 (8 is not square free)
24 = 4 * 6 (4 is not square free)
24 = 6 * 2 * 2 (Every divisor is square free with divisor count = 3)
24 = 3 * 2 * 2 * 2 (Every divisor is square free with divisor count = 4)
Hence, appropriate answer would be 3 (24 = 6 * 2 * 2). Now if we observe carefully, we can’t group a prime factor with itself but with another prime factor. Like in this case, 2 is grouped with 3 to make 6 (square free) to reduce divisors count. Thus, answer would be maximum of maximum powers of all prime factors because we need atleast that much count to retain the condition of square free and other prime factors (whose power is not maximum) can be grouped with the prime factor with maximum power to minimize the count of divisors.
Below is the implementation of above approach. Here, we will do preprocessing of finding prime factors (using Sieve) so that we can answer many queries simultaneously.
Minimum Number of Square Free Divisors is 3 Minimum Number of Square Free Divisors is 1
- Square Free Number
- Nth Square free number
- Break a number such that sum of maximum divisors of all parts is minimum
- Find minimum number to be divided to make a number a perfect square
- Divisors of n-square that are not divisors of n
- Minimum digits to remove to make a number Perfect Square
- Minimum number of square tiles required to fill the rectangular floor
- Find sum of inverse of the divisors when sum of divisors and the number is given
- Find sum of divisors of all the divisors of a natural number
- Check if a number is perfect square without finding square root
- Querying maximum number of divisors that a number in a given range has
- Find the number of integers x in range (1,N) for which x and x+1 have same number of divisors
- Find the largest good number in the divisors of given number N
- Check if a number is divisible by all prime divisors of another number
- Find the sum of the number of divisors
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.