Lecture 3: GCD and LCM - Math
GCF vs. LCM. The Greatest Common Factor (or the GCF) is the greatest real number shared between two integers. What makes this number a. Definition: The greatest common divisor of two integers a and b, written, is the largest integer d so that and. More generally, if . Our goal, then, is to show that. For this .. The Relationship between LCM and GCD. One of the. Answer to An interesting relationship between the GCD and LCM of two numbers is that the product of the two numbers is equal to.
And we could keep going on and on in there. But let's see what they're asking us.GCD and LCM
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.
GCF & LCM word problems (video) | Khan Academy
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. 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. For obvious reasons, such a number is called a common divisor.
Difference Between GCF and LCM
Certainly common divisors exist for any pair of integers a and b, since we know that 1 always divides any integer. With all this as motivation, we have the following Definition: Non-trivial GCD Suppose we'd like to know the greatest common divisor of 12 and Trivial GCD If we want to know the greatest common divisor of 21 and 10, then we write down their divisors: This follows since 0 has the property that every integer divides it.
Notice in this last example that the collection is relatively prime even though each pair of integers from the collection is not relatively prime.
As a general rule of thumb, you'll care more about whether a collection is pairwise relatively prime than whether it's relatively prime.
GCF & LCM word problems
Properties of the GCD Having met and played around with greatest common divisors a bit, we'll now introduce a few properties that they enjoy. Removing the GCD First, we'll see what we get when we remove the gcd of two integers.
Toward this end, we'll start by showing that d is a common divisor of both a and b, then we'll show that all other common divisors divide d and so all other divisors are no bigger than d.
So d is a common divisor. So would a 2-by-2 or a 3-by-3 square. Notice that these numbers are all common factors of 24 and To determine the GCF, we want to find the dimensions of the largest square that could tile the entire rectangle without gaps or overlap. Here's one quick method. Start with the by rectangle: The largest square tile that fits inside this rectangle and is flush against one side is 24 by Only one tile of this size will fit: The largest square tile that fits inside the remaining rectangle and is flush against one side is 12 by Two tiles of this size will fit.