How to Get Numbers from Given Gcd and Lcm

How to get GCD and LCM of a range of numbers efficiently?

As mentioned by Chris J this SO question provides the algorithm. Here is a python version of the answer that uses the reduce built-in and the fractions module that has been around since version 2.6.

import fractions

def gcd(*values):
    return reduce(fractions.gcd, values)

How to find GCD, LCM on a set of numbers

I've used Euclid's algorithm to find the greatest common divisor of two numbers; it can be iterated to obtain the GCD of a larger set of numbers.

private static long gcd(long a, long b)
{
    while (b > 0)
    {
        long temp = b;
        b = a % b; // % is remainder
        a = temp;
    }
    return a;
}

private static long gcd(long[] input)
{
    long result = input[0];
    for(int i = 1; i < input.length; i++) result = gcd(result, input[i]);
    return result;
}

Least common multiple is a little trickier, but probably the best approach is reduction by the GCD, which can be similarly iterated:

private static long lcm(long a, long b)
{
    return a * (b / gcd(a, b));
}

private static long lcm(long[] input)
{
    long result = input[0];
    for(int i = 1; i < input.length; i++) result = lcm(result, input[i]);
    return result;
}

GCD and LCM relation

The analogous formulas to

LCM(a, b) = (a x b) / GCD(a,b) or GCD(a,b) = (a x b) / LCM(a, b) 

with three variables are simply not valid, as your example with (3, 12, 10) shows readily.

The product of these three numbers is 360. The GCD is 1. The LCM is 60.

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