1808. Maximize Number of Nice Divisors

Description

You are given a positive integer primeFactors. You are asked to construct a positive integer n that satisfies the following conditions:

• The number of prime factors of n (not necessarily distinct) is at most primeFactors.
• The number of nice divisors of n is maximized. Note that a divisor of n is nice if it is divisible by every prime factor of n. For example, if n = 12, then its prime factors are [2,2,3], then 6 and 12 are nice divisors, while 3 and 4 are not.

Return the number of nice divisors of n. Since that number can be too large, return it modulo 109 + 7.

Note that a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. The prime factors of a number n is a list of prime numbers such that their product equals n.

 

Example 1:

Input: primeFactors = 5
Output: 6
Explanation: 200 is a valid value of n.
It has 5 prime factors: [2,2,2,5,5], and it has 6 nice divisors: [10,20,40,50,100,200].
There is not other value of n that has at most 5 prime factors and more nice divisors.

Example 2:

Input: primeFactors = 8
Output: 18

 

Constraints:

• 1 <= primeFactors <= 109

 

Solutions

Solution: Math

• Time complexity: O(logn)
• Space complexity: O(1)

 

JavaScript

/**
 * @param {number} primeFactors
 * @return {number}
 */
const maxNiceDivisors = function (primeFactors) {
  if (primeFactors <= 3) return primeFactors;
  const MODULO = BigInt(10 ** 9 + 7);

  const powMod = (base, exponent) => {
    let result = 1n;

    while (exponent) {
      if (exponent % 2n) {
        result = (result * base) % MODULO;
      }

      base = (base * base) % MODULO;
      exponent /= 2n;
    }

    return result;
  };

  const count = BigInt(Math.floor(primeFactors / 3));
  const remainder = primeFactors % 3;

  if (remainder === 0) return Number(powMod(3n, count));
  if (remainder === 1) {
    const result = powMod(3n, count - 1n);

    return Number((result * 4n) % MODULO);
  }

  return Number((powMod(3n, count) * 2n) % MODULO);
};