703973. Powers game
Here is a detailed Markdown blog post for the "Powers game" problem:
Powers Game
Summary
The powers game is an algorithmic problem that involves calculating the maximum sum of powers of a given number. The problem requires understanding of modular arithmetic and logical thinking. Given a positive integer n, find the maximum value of n^k modulo 10^9 + 7 for all 1 <= k <= n.
Detailed Explanation
The key to solving this problem is to understand how powers of numbers behave when taken modulo a large number. One way to approach this problem is by using modular arithmetic properties.
Here's the step-by-step breakdown of the solution:
- Initialize the maximum sum to 0.
- Iterate over all values of
kfrom 1 ton. - For each
k, calculaten^kmodulo(10^9 + 7). This can be done using the property that(a * b) mod c = ((a mod c) * (b mod c)) mod c. - Add the calculated value to the maximum sum.
- Return the maximum sum.
Here's an ASCII art diagram illustrating the steps:
1. Initialize max_sum
2. Iterate over k from 1 to n
3. Calculate n^k mod (10^9 + 7)
-> ((n mod (10^9 + 7)) * (k mod (10^9 + 7))) mod (10^9 + 7)
4. Add calculated value to max_sum
5. Return max_sum
The time complexity of this solution is O(n), where n is the input size, and the space complexity is O(1) as we only use a constant amount of space.
Optimized Solutions
Java
public int maxPowers(int n) {
long maxSum = 0;
for (int k = 1; k <= n; k++) {
maxSum += pow(n, k, 1000000007);
}
return (int)maxSum;
}
private long pow(long a, long b, long mod) {
long res = 1;
while (b > 0) {
if ((b & 1) == 1)
res = (res * a) % mod;
a = (a * a) % mod;
b >>= 1;
}
return res;
}