### Greatest Common Factor | Least Common Multiple | [email protected]

The other names for GCD are Greatest Common Divisor(GCD), Highest common factor and Greatest Common denominator. GCD is Relationship between LCM and GCD is given as: The multiplies of 6 are 6, 12, 18, 24, 30, 36, 42,. The common factors of 18 and 30 are 1, 2, 3 and 6. The highest Solution. a: Because 6 is a factor of 24, the HCF of 6 and 24 is 6. Two relationships between the HCF and LCM. The two. Using the language of mathematics, we say that 6, 12, 18 and 24 are multiples of 6. In fact, all . equation summarises the relationship between two numbers ('. factor .. 6. What are the highest common factors of the pairs of numbers below?.

But the product of the numbers is not necessarily their lowest common multiple. What is the general situation illustrated here? Solution The LCM of 9 and 10 is their product The common multiples are the multiples of the LCM You will have noticed that the list of common multiples of 4 and 6 is actually a list of multiples of their LCM Similarly, the list of common multiples of 12 and 16 is a list of the multiples of their LCM This is a general result, which in Year 7 is best demonstrated by examples.

In an exercise at the end of the module, Primes and Prime Factorisationhowever, we have indicated how to prove the result using prime factorisation. This can be restated in terms of the multiples of the previous section: On the other hand, zero is the only multiple of zero, so zero is a factor of no numbers except zero. These rather odd remarks are better left unsaid, unless students insist. They should certainly not become a distraction from the nonzero whole numbers that we want to discuss.

The product of two nonzero whole numbers is always greater than or equal to each factor in the product. Hence the factors of a nonzero number like 12 are all less than or equal to Thus whereas a positive whole number has infinitely many multiples, it has only finitely many factors. The long way to find all the factors of 12 is to test systematically all the whole numbers less than 12 to see whether or not they go into 12 without remainder.

And if there's a sixth test, then they would get to And we could keep going on and on in there.

But let's see what they're asking us. What is the minimum number of exam questions William's or Luis's class can expect to get in a year? Well the minimum number is the point at which they've gotten the same number of exam questions, despite the fact that the tests had a different number of items. And you see the point at which they have the same number is at This happens at They both could have exactly questions even though Luis's teacher is giving 30 at a time and even though William's teacher is giving 24 at a time.

And so the answer is And notice, they had a different number of exams. Luis had one, two, three, four exams while William would have to have one, two, three, four, five exams. But that gets them both to total questions.

Now thinking of it in terms of some of the math notation or the least common multiple notation we've seen before, this is really asking us what is the least common multiple of 30 and And that least common multiple is equal to 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.

### Multiples, Factors and Powers

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.

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

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.

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

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.

### GCF & LCM word problems (video) | Khan Academy

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. 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? So we could go with the prime factor. We could list all of their normal factors and see what is the greatest common one. Or we could look at the prime factorization.