Question

How to find the lowest common multiple?

Answer 1

Break down each number in your list into its prime factors, expressed as a product of primes raised to various powers, then take product of the power with the greatest exponent in each prime.

Explanation:

Recall that the lowest common multiple of a list of numbers must be divisible by all of those numbers.

One method is to break down a number into a product of prime factors. Consider the LCM of 4, 6, 7, and 8.

`4 = 2 * 2`
`4 = 2^2`

`6 = 2 * 3`

`7 = 7`

`8 = 2 * 2 * 2`
`8 = 2^3`

In order for our LCM to be divisible by all of these numbers, for each number, the prime product of LCM must contain all of the number's prime factors.

To minimize this number, we will only include necessary primes in our product. For example, we only need to include `2^3` in order to have a number divisible by both 4 and 8 as 4's prime factors, `2^2`, are already included.

Simply put, we need to take the greatest exponent of each of our prime numbers in the prime products of our numbers.

The prime `2` is included in:
- `4` as `2^2`
- `6` as `2`
- `8` as `2^3`
Greatest is `2^3`.

The prime `3` is included in:
- `6` as `3`
Greatest is `3`.

The prime `7` is included in:
- `7` as `7`
Greatest is `7`.

So, the minimum number divisible by 4, 6, 7, and 8 can be represented as

`2^3*3*7`

or calculated, as

`168`