Multiples, factors, and factorizations are three of the easiest building blocks of mathematics, and most people are familiar with that. However, if you don’t know the terms, here’s a quick introduction to the topic of the article that we need to know.
Firstly, a fundamental concept of math is multiplication, and it results in multiples. However, let’s demystify something before introducing more of its features.
Is 4.5 a multiple of 3 because it is 3 * 1.5? Absolutely not! Multiples are only suitable for integers, not decimal numbers. If the opposite is true, the concept of “multiples” is broken. Any number could be a “multiple” for any number if that happened.
Therefore, the first multiple for 6 is 6 (6 * 1), the second is 12 (6 * 2), the third is 18 (6 * 3), and so on. Just multiply the subject with the variable, and the variableth multiple of the subject is present.
Moreover, you can multiply a number with a negative one, too. But, keep in mind that if a positive number is multiplied by a negative number, the result is negative. Otherwise, it’s positive.
Factors are numbers that are divisible by the subject. Remember to include the subject itself, too, as 1 is a factor of every integer. For instance, 1, 2, 3, and 6 are factors of 6 because 1, 2, 3, and 6 are all divisible by 6. Moreover, 1, 2, 4, and 8 are factors of 8 due to the same reason (only the subject is modified).
There are many types of numbers that are associated with factors. For example, perfect numbers are equal to half the sum of all of its factors. For instance, 28 is a perfect number because its sum of factors is 1 + 2 + 4 + 7 + 8 + 14 + 28 = 56, twice the amount of 28.
Prime numbers are a fundamental type of number, too. Every prime number has only two factors: 1 and itself, which is the bare minimum. For instance, 7 can’t be divided by 2, 3, 4, 5, or 6, and only 1 and 7 are divisible by 7. Therefore, 7 is a prime number.
Interestingly, prime numbers are an essential element of factorizations, which will be explained in the next section.
Before you find out the factors of a number, especially for a complex one that is, for example, a 5-digit or a 7-digit one, you have to factorize it. It is a verb to express the process of extracting its prime factors.
Let’s take an example from a smaller integer like 12. You know that 2 and 3 are prime numbers, right? Then, divide 12 by 2, and you’ll get 6. Divide it by 2 again because it’s an even number. Finally, we get 2 * 2 * 3, or 2 ^ 2 * 3.
Have you ever wondered how to count how many factors are there for a 6-digit number? It isn’t dividing every single number before it and see if what matches, right? Fortunately, factorization is a speedy method to get the value.
Let’s take 12 as an example again. Remember that it is 2 ^ 2 * 3, and we use exponentiation for counting the count of factors. Then, the “^2” serves us with three (2 + 1) factors. The “3” provides us with two factors because it is in the first power.
Finally, we multiply the factor each prime number brings in. 2 * 3 equals 6, so 12 has 6 factors.
The Least Common Multiple (LCM)
Now, let’s talk about more basic things you can do with multiples, factors, and factorizations.
Important: Don’t mix LCM and LMC (Large Magellanic Cloud) up.
To understand LCM, we need to learn what is a “common multiple”. A common multiple is a multiple of all parameters provided. For example, a common multiple of 3, 4, and 5 is 120.
However, LCM is about finding the least of the common multiples, which contains infinitely many items like regular multiples. Therefore, if we take the number 120, it is not the LCM since 60 is a common multiple of 3, 4, and 5, too.
Then, how to find them? We can either list all multiples of all items and see if something is overlapping on all lists, or use this formula, which involves the GCF. However, it’s only correct for the LCM of two items.
Note: The GCF function here can also be HCF or GCD. We’ll talk about that in the next section.
The Greatest Common Factor (GCF)
The greatest common factor (GCF) (a.k.a. HCF or GCD) is self-explanatory. Therefore, let’s go right into the example.
For instance, the GCF of 8 and 14 is 2. Let’s list their factors:
8: 1, 2, 4, 8
14: 1, 2, 7, 14
This is the easiest way to find the GCFs of multiple numbers. However, if it’s a large number like a 5-digit one, you may want to start small. For example, if both are even numbers, divide them by two and find the GCF on the results. Once you’ve found all common factors, just insert the answer.
Here is a brief introduction to multiples, factors, and factorizations in this article. However, this article is not enough. As a result, please look at the references to keep exploring the topics.
References and Credits
- (n.d.). Factors and Multiples – Math is Fun. Retrieved January 23, 2021, from https://www.mathsisfun.com/numbers/factors-multiples.html
- (n.d.). Least Common Multiple – Math is Fun. Retrieved January 23, 2021, from https://www.mathsisfun.com/least-common-multiple.html
- (n.d.). Perfect Number — from Wolfram MathWorld. Retrieved January 23, 2021, from https://mathworld.wolfram.com/PerfectNumber.html
- (n.d.). Perfect Number – Arizona Math – University of Arizona. Retrieved January 23, 2021, from https://math.arizona.edu/~rta/001/gaberdiel/
- (n.d.). Greatest Common Factor – Math is Fun. Retrieved January 23, 2021, from https://www.mathsisfun.com/greatest-common-factor.html
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