### Math Forum - Ask Dr. Math Archives: LCM, GCF

The common factors of 18 and 30 are 1, 2, 3 and 6. The highest common b: The only factor of 1 is 1, so the HCF of 1 and 14 is 1. Two relationships between the HCF and LCM. The two. GCF of 14 and 18, find the biggest number that can divide two integers, calculate factors and the greatest common factor of 14 and Least Common Multiple (LCM), Greatest Common Factor (GCF). A common multiple is a Common multiples of 2 and 3 are 0, 6, 12, 18, The least common.

Therefore, if we were asked to list factors of 10, we would write: List the factor pairs and factor lists of the following numbers: Notice that when we wrote our lists, we started by writing the number given to us originally, followed by a colon: This is standard notation, but not absolutely necessary.

Next, we wrote factor pairs. Factor pairs are always the multiplication problems, and look like this: Last, we listed the factors themselves, separated by commas. You may be asked for just the factor pairs, or just the factor list, or both.

- Prime Factorization of 18
- Prime Factorization of 14
- Greatest Common Factor

Greatest Common Factor After you learn how to find factors, you may be asked to find the greatest common factor GCF between two numbers. The GCF is the largest factor that both numbers share.

## Relationship between H.C.F. and L.C.M.

This means that you would list all the factors for each number. Then, you would circle or underline, etc all the factors that the numbers have in common. After that, you would report the greatest number out of the common—circled—numbers. For example, find the greatest common factor of 24 and Suppose that each number in the table is divided by 7 to produced a quotient and a remainder. What is the same about the results of the division in each row?

Common multiples and the LCM An important way to compare two numbers is to compare their lists of multiples.

### Greatest common factor examples (video) | Khan Academy

Let us write out the first few multiples of 4, and the first few multiples of 6, and compare the two lists. The numbers that occur on both lists have been circled, and are called common multiples. The common multiples of 6 and 8 are 0, 12, 24, 36, 48,… Apart from zero, which is a common multiple of any two numbers, the lowest common multiple of 4 and 6 is These same procedures can be done with any set of two or more non-zero whole numbers. A common multiple of two or more nonzero whole numbers is a whole number that a multiple of all of them.

The lowest common multiple or LCM of two or more whole numbers is the smallest of their common multiples, apart from zero. Hence write out the first few common multiples of 12 and 16, and state their lowest common multiple. Now there's other ways that you can find the least common multiple other than just looking at the multiples like this.

You could look at it through prime factorization. So we could say that 30 is equal to 2 times 3 times 5. And that's a different color than that blue-- 24 is equal to 2 times So 24 is equal to 2 times 2 times 2 times 3. So another way to come up with the least common multiple, if we didn't even do this exercise up here, says, look, the number has to be divisible by both 30 and If it's going to be divisible by 30, it's going to have to have 2 times 3 times 5 in its prime factorization.

That is essentially So this makes it divisible by And say, well in order to be divisible by 24, its prime factorization is going to need 3 twos and a 3. Well we already have 1 three. And we already have 1 two, so we just need 2 more twos. So 2 times 2. So this makes it-- let me scroll up a little bit-- this right over here makes it divisible by And so this is essentially the prime factorization of the least common multiple of 30 and You take any one of these numbers away, you are no longer going to be divisible by one of these two numbers.

If you take a two away, you're not going to be divisible by 24 anymore.

**Greatest Common Factor 18 and 12 TEK 6.1E**

If you take a two or a three away. If you take a three or a five away, you're not going to be divisible by 30 anymore.

And so if you were to multiply all these out, this is 2 times 2 times 2 is 8 times 3 is 24 times 5 is Now let's do one more of these. Umama just bought one package of 21 binders. Let me write that number down.

She also bought a package of 30 pencils. She wants to use all of the binders and pencils to create identical sets of office supplies for her classmates. What is the greatest number of identical sets Umama can make using all the supplies? So the fact that we're talking about greatest is clue that it's probably going to be dealing with greatest common divisors. And it's also dealing with dividing these things. We want to divide these both into the greatest number of identical sets.

So there's a couple of ways we could think about it.

## Greatest common factor examples

Let's think about what the greatest common divisor of both these numbers are. Or I could even say the greatest common factor. The greatest common divisor of 21 and So what's the largest number that divides into both of them?