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Our least common factor calculator will help you with all your math problems: in the blink of an eye, we will tell you both what mathematics tells us is the least common factor and also the first multiple common to all numbers of a set of (almost) any size, that is, in fact, the least common multiple. Keep reading this article to learn:

  • What is the least common factor compared to the least common multiple;
  • How to calculate the least common factor;
  • If you need to calculate the least common multiple using the greatest common divisor;
  • How to use prime factorization for the least common multiple calculations;
  • How to find the least common multiple: other methods.

And much more!

What is the least common factor?

The least common factor of a set of numbers is the smallest non-trivial prime factor shared by a set of numbers. Why do we need the least common factor? The least common factor allows us to identify coprime numbers: pairs, or sets, of divisible numbers only by 11. Coprime numbers have some interesting mathematical properties. In engineering, we use them to create sets of gears where every tooth of each gear touches every tooth of contacting gears for an equal amount of time (there are no "preferred" pairs that share more time in contact).

Every set of numbers you can imagine has trivial least common factor 11. However, as 11 is not considered to be a prime number, we can ignore it in our computations.

The difference between least common factor and least common multiple

However, detecting coprimes is probably not the reason you are here in the first place! In the following sections, you will learn how to calculate the least common multiple!

On the other hand, the least common multiple is the smallest integer that is multiple to all numbers in the set. The multiple of a number is the result of the multiplication of said number by any integer larger than 11 (or 00, but it's... debatable). This means that for each set of numbers you can imagine, you can identify a unique combination of integer numbers that, multiplied by the number associated with each, return the same result. If the greatest common divisor of this combination is 11, then the result is the least common factor.

How to calculate the least common factor: how to find if two numbers are not coprimes

You are in the right place if you need to know how to find the least common factor. Follow these easy steps to find the smallest non-trivial prime factor shared by a pair of numbers.

  1. Take the pair of numbers aa and bb.
  2. Compute the prime factorization of aa.
  3. Compute the prime factorization of bb.
  4. Starting from the smallest factors of both numbers, check if you find a correspondence.
    • If both numbers share the first factor, you successfully calculate the least common factor, and the two numbers are , not coprimes.
  5. Take the largest of the two factors, and compare it with the factors of the other number until you find a correspondence (hence you found the least common factor) or a factor larger than the one you are considering. In this case, use the larger factor just found to run the same comparison of this step.
  6. If you reach the end of both factorizations without finding a common factor, the two numbers are coprimes.

Now that you learned how to find the least common factor, let's learn how to calculate the

How to ignore prime factorization and calculate the least common multiple using the greatest common divisor

The least common multiple and the greatest common divisor(GCD) are different concepts but share many similarities: they both are related to a set of numbers. They are associated with fundamental quantities like multiples and divisors. Unsurprisingly, we can find one with the help of the other.

While we don't usually use the least common multiple to find the greatest common divisor, as we have efficient algorithms, finding the least common multiple from the GCD is common practice. The formula we use for this is pretty straightforward. For a set {a,b}\{a,b\}, the least common multiple is given by:

lcm(a,b)=abgcd(a,b)\mathrm{lcm}(a,b) = \frac{\big|a\cdot b\big|}{\mathrm{gcd}(a,b)}

If the set contains more than 2 numbers, once you find the least common multiple for a pair of them, keep applying the algorithm substituting each pair of numbers with the relative least common multiple.

Calculate the least common multiple using prime factorization

To find the least common multiple using prime factorization, we must first learn what prime factorization is! We say that we found the prime factors of a number when we identify the prime numbers that multiplied returns the number itself. It may be counterintuitive, but we can only build every possible number using prime factors.

In the case of numbers where a certain prime factor repeats (like 12=22312 = 2\cdot 2 \cdot3), we can group the repeated factors using powers. This property will come in handy!

To find the least common multiple of a set of numbers using only prime factorization, write down all the factors of the numbers using powers, and select all the unique characteristics with the highest possible exponent. Find their product to find the least common multiple.

Other methods to find the least common mutliple

You can find the least common multiple in other math-free methods:

  • The table method; and
  • The list of multiples method.

In the former, we divide all numbers in a set by growing prime numbers starting with 22. If some numbers are divisible by 22, we substitute the result in the set. Repeat the division by two until no number in the set is divisible, then move on to 33 and repeat the steps above. Whenever at least a number in the set is divisible by a prime, mark it down. The product of such numbers gives you the least common mutliple.

If you want to list multiples, start by writing down an arbitrary number of them for the first number, say, 10. Then, move to the following number. If you find a common multiple, you're good to go; if not, add more multiples. If you have more than one number, once you find the least common multiple for the first two numbers, consider the pair formed by the first least common multiple found and the last number, and list their multiples.

Other least common factor calculators

Keep exploring this mathematical quantity with the following:

FAQ

What is the least common multiple of 3, 7, and 25?

The least common multiple of 8, 12, and 25 is 600. To find this result, use the following steps:

  1. Calculate the product of the first two numbers: 8 × 12 = 96.
  2. Compute their gcd: gcd(12,8) = 4.
  3. Divide the product by the gcd: this is the least common multiple of 8 and 12: lcm(8,12) = 96/4 = 24.
  4. Compute the product and gcd of 25 and the result from step 3.:
    • 25 × 24 = 600;
    • gcd(25,24) = 1.
  5. Find the lcm: lcm = 600/1 = 600.

The result is then lcm(8,12,25) = 600.

How do I find the least common multiple using prime factors?

To find the least common multiple using prime factors, follow these easy steps:

  1. Find the prime factorization of all numbers in the set. If factors repeat in a factorization, use powers to write them.
  2. Group together all the factors and keep all the unique factors. If factors repeat, consider the ones with the highest exponent.
  3. Multiply the factors you selected to find the least common multiple.

How do I find the least common multiple of 12, 16, and 18?

List all the multiples of these three numbers to find the least common multiple of 12, 16, and 18 until you find a common match. We can do this pairwise:

  1. The multiples of 12 are 12, 24, 36, 48, 60, 72, 84,...
  2. The multiples of 16 are 16, 32, 48,...
  3. The lcm of 12 and 16 is 48.
  4. List the multiples of 48 and 18:
    • 48: 48, 96, 144, 192, 240,..
    • 18: 18, 36, 54, 72, 90, 108, 126, 144,..

That's it: we found the second match: lcm(12,16,18) = 144.

Davide Borchia
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