Here is a detailed markdown blog post for the LCM problem:

LCM

Summary

This problem deals with calculating the Least Common Multiple (LCM) of two given numbers. The key concept involved is finding the smallest positive integer that is divisible by both numbers.

Detailed Explanation

To solve this problem, we can use the mathematical formula for LCM, which is the product of each distinct prime factor of both numbers raised to the power of the highest power it appears in either number.

Here's a step-by-step breakdown:

Find the prime factors of both numbers using any prime factorization algorithm.
For each distinct prime factor, find the maximum power it appears in either number.
Calculate the LCM by multiplying the prime factor to the power of its highest appearance in either number.

Time complexity: O(sqrt(max(a, b))), where a and b are the input numbers. This is because we need to iterate up to the square root of the maximum number to find all its prime factors.

Space complexity: O(1), as we only need a constant amount of space to store the LCM result.

Optimized Solutions

Java

public int findLCM(int a, int b) { int max = Math.max(a, b); for (int i = 2; i * i <= max; i++) { while (max % i == 0) { max /= i; } if (i > 1 && a % i != 0 && b % i != 0) { return i; } } return max; }