The Master Key To Solving Puzzles, Part 3

Mind On The Puzzle

The Master Key To Solving Puzzles, Part 2

Tenets (Part 3)
1. Tool Your Solution With Mathematics
2. Walking Through Scenarios

Tenet: Tool Your Solution With Mathematics Walking Through Scenarios

The art of thought is philosophy; the mechanics of thought is science; the language of thought is mathematics.

[Synching Break-Time Riddle]
You and your friend have 1 unit of task to complete. You can complete 1 unit of task in 8 hours, and your friend can complete it in 12 hours. How long will it take for both of you to complete the task at the same time to go for a break together? 

Answer: 24 hours.
The LCM of 2 or more numbers is the smallest positive integer that is divisible by all of them.

The LCM of 8 and 12 is 24, which represents the least amount of time in which both of you can finish the task batch units at the same time without each starting the next batch) 
8     12
16   [24]
[24]  36

The greatest common factor (GCF) is a term used to describe the biggest number that can divide evenly into two or more numbers. Sometimes, this is also referred to as the greatest common divisor (GCD) or highest common factor (HCF).

The GCD of 2 or more numbers is the largest positive integer that divides all of them. GCD is used in various mathematical problems, such as finding the HCF of 2 or more numbers, solving Diophantine equations, and reducing fractions to their lowest terms.

GCD(8, 12) is 4.
8    12 (÷ 4) 
   ⥮
2     3 (x 4) 

This is where GCD(2, 3) = 1, i.e. we find the basis where they are relative prime of each other.

Cross-multiplying,
8     12 
   X
2      3   
8x3 = 12x2

The Least Common Multiple (LCM) and Greatest Common Divisor (GCD) are mathematical concepts that have different purposes and applications.

LCM and GCD is also used to find the common denominator given a list of fractions. 
e.g. 3, 7, 15, 31, 63, the LCM = 9765. 

"""
import math
import numpy as np

def __gcd(a, b):
    if (a == 0):
        return b
    return __gcd(b % a, a)
 
# recursive implementation
def lcm_of_array(arr, idx=0):
    # lcm(a,b) = (a*b/gcd(a,b))
    if (idx == len(arr)-1):
        return arr[idx]
    a = arr[idx]
    b = lcm_of_array(arr, idx+1)
    return…