# Posts tagged as “math”

Given an integer n (in base 10) and a base k, return the sum of the digits of n after converting n from base 10 to base k.

After converting, each digit should be interpreted as a base 10 number, and the sum should be returned in base 10.

Example 1:

Input: n = 34, k = 6
Output: 9
Explanation: 34 (base 10) expressed in base 6 is 54. 5 + 4 = 9.


Example 2:

Input: n = 10, k = 10
Output: 1
Explanation: n is already in base 10. 1 + 0 = 1.


Constraints:

• 1 <= n <= 100
• 2 <= k <= 10

## Solution: Base Conversion

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

## C++

You are given a string s (0-indexed)​​​​​​. You are asked to perform the following operation on s​​​​​​ until you get a sorted string:

1. Find the largest index i such that 1 <= i < s.length and s[i] < s[i - 1].
2. Find the largest index j such that i <= j < s.length and s[k] < s[i - 1] for all the possible values of k in the range [i, j] inclusive.
3. Swap the two characters at indices i - 1​​​​ and j​​​​​.
4. Reverse the suffix starting at index i​​​​​​.

Return the number of operations needed to make the string sorted. Since the answer can be too large, return it modulo 109 + 7.

Example 1:

Input: s = "cba"
Output: 5
Explanation: The simulation goes as follows:
Operation 1: i=2, j=2. Swap s and s to get s="cab", then reverse the suffix starting at 2. Now, s="cab".
Operation 2: i=1, j=2. Swap s and s to get s="bac", then reverse the suffix starting at 1. Now, s="bca".
Operation 3: i=2, j=2. Swap s and s to get s="bac", then reverse the suffix starting at 2. Now, s="bac".
Operation 4: i=1, j=1. Swap s and s to get s="abc", then reverse the suffix starting at 1. Now, s="acb".
Operation 5: i=2, j=2. Swap s and s to get s="abc", then reverse the suffix starting at 2. Now, s="abc".


Example 2:

Input: s = "aabaa"
Output: 2
Explanation: The simulation goes as follows:
Operation 1: i=3, j=4. Swap s and s to get s="aaaab", then reverse the substring starting at 3. Now, s="aaaba".
Operation 2: i=4, j=4. Swap s and s to get s="aaaab", then reverse the substring starting at 4. Now, s="aaaab".


Example 3:

Input: s = "cdbea"
Output: 63

Example 4:

Input: s = "leetcodeleetcodeleetcode"
Output: 982157772


Constraints:

• 1 <= s.length <= 3000
• s​​​​​​ consists only of lowercase English letters.

## Solution: Math

Time complexity: O(26n)
Space complexity: O(n)

## C++

There are n friends that are playing a game. The friends are sitting in a circle and are numbered from 1 to n in clockwise order. More formally, moving clockwise from the ith friend brings you to the (i+1)th friend for 1 <= i < n, and moving clockwise from the nth friend brings you to the 1st friend.

The rules of the game are as follows:

1. Start at the 1st friend.
2. Count the next k friends in the clockwise direction including the friend you started at. The counting wraps around the circle and may count some friends more than once.
3. The last friend you counted leaves the circle and loses the game.
4. If there is still more than one friend in the circle, go back to step 2 starting from the friend immediately clockwise of the friend who just lost and repeat.
5. Else, the last friend in the circle wins the game.

Given the number of friends, n, and an integer k, return the winner of the game.

Example 1:

Input: n = 5, k = 2
Output: 3
Explanation: Here are the steps of the game:
1) Start at friend 1.
2) Count 2 friends clockwise, which are friends 1 and 2.
3) Friend 2 leaves the circle. Next start is friend 3.
4) Count 2 friends clockwise, which are friends 3 and 4.
5) Friend 4 leaves the circle. Next start is friend 5.
6) Count 2 friends clockwise, which are friends 5 and 1.
7) Friend 1 leaves the circle. Next start is friend 3.
8) Count 2 friends clockwise, which are friends 3 and 5.
9) Friend 5 leaves the circle. Only friend 3 is left, so they are the winner.

Example 2:

Input: n = 6, k = 5
Output: 1
Explanation: The friends leave in this order: 5, 4, 6, 2, 3. The winner is friend 1.


Constraints:

• 1 <= k <= n <= 500

## Solution 1: Simulation w/ Queue/ List

Time complexity: O(n*k)
Space complexity: O(n)

## C++/List

There is a function signFunc(x) that returns:

• 1 if x is positive.
• -1 if x is negative.
• 0 if x is equal to 0.

You are given an integer array nums. Let product be the product of all values in the array nums.

Return signFunc(product).

Example 1:

Input: nums = [-1,-2,-3,-4,3,2,1]
Output: 1
Explanation: The product of all values in the array is 144, and signFunc(144) = 1


Example 2:

Input: nums = [1,5,0,2,-3]
Output: 0
Explanation: The product of all values in the array is 0, and signFunc(0) = 0


Example 3:

Input: nums = [-1,1,-1,1,-1]
Output: -1
Explanation: The product of all values in the array is -1, and signFunc(-1) = -1


Constraints:

• 1 <= nums.length <= 1000
• -100 <= nums[i] <= 100

## Solution: Sign Only

No need to compute the product, only track the sign changes. Flip the sign when encounter a negative number, return 0 if there is any 0 in the array.

Time complexity: O(n)
Space complexity: O(1)

## C++

You are given an array nums that consists of positive integers.

The GCD of a sequence of numbers is defined as the greatest integer that divides all the numbers in the sequence evenly.

• For example, the GCD of the sequence [4,6,16] is 2.

subsequence of an array is a sequence that can be formed by removing some elements (possibly none) of the array.

• For example, [2,5,10] is a subsequence of [1,2,1,2,4,1,5,10].

Return the number of different GCDs among all non-empty subsequences of nums.

Example 1:

Input: nums = [6,10,3]
Output: 5
Explanation: The figure shows all the non-empty subsequences and their GCDs.
The different GCDs are 6, 10, 3, 2, and 1.


Example 2:

Input: nums = [5,15,40,5,6]
Output: 7


Constraints:

• 1 <= nums.length <= 105
• 1 <= nums[i] <= 2 * 105

## Solution: Math

Enumerate all possible gcds (1 to max(nums)), and check whether there is a subset of the numbers that can form a given gcd i.
If we want to check whether 10 is a valid gcd, we found all multipliers of 10 in the array and compute their gcd.
ex1 gcd(10, 20, 30) = 10, true
ex2 gcd(20, 40, 80) = 20, false

Time complexity: O(mlogm)
Space complexity: O(m)