最小公倍数

Finding the Least Common Multiple

The least common multiple is exactly what it sounds like - it's the smallest quantity that is the multiple of two or more other quantities.

Here's a simple example: Let's say we want to know the least common multiple of $3$ and $4$ (remember, these are both degree zero polynomials) This means that we want to know the smallest number that is a multiple of both $3$ and $4.$ Another way to think about the LCM is as the smallest number that can be evenly divided by both $3$ and $4.$

In this example, the LCM is $12.$ This is the smallest possible number that is a multiple of $3$ and $4.$

As TenMarks Amazon mentioned in his video, there are 3 steps to follow when finding the LCM of two or more terms:

1. Factor both terms

2. Identify common factors

3. Multiply unique factors and the common factors with the highest exponents

Let's take a look at these steps in action.

Consider these two terms: $10a^4b^3$ and $12ab^5$

STEP 1: Factor both terms.

Remember that, to factor monomials, we want to break it apart or expand it. This involves breaking the coefficient down into prime numbers.

For instance, when we factor $10a^4b^3$, we get $2\cdot5\cdot a^4\cdot b^3$

When we expand $12ab^5$ and break the coefficient ($12$) into prime numbers, we get $2\cdot2\cdot3\cdot a\cdot b^5$.

So, here are our two factored terms:

STEP 2: Identify Common Factors

STEP 3: Multiply unique factors and the common factors with the highest exponents

What factors are unique - meaning that they are not shared by both terms?

In this last step, we're going to multiply the unique factors and the common factors with the highest exponents.

So, let's multiply the unique terms ($3$ and $5$) by the common factors with the highest exponents.

That looks like this: $3\cdot5\cdot2^2\cdot a^4\cdot b^5$

This means that our LCM is the product of these terms, or $60a^4b^5$

(Videos) Techniques to Find the LCM

by PatrickJMT and eHow

Here's a video by PatrickJMT explaining what the least common multiple is and how to find it. The least common multiple is the smallest number that has each of the numbers as a factor. The steps he uses for both numbers and variables are:

1. Write the prime factorization of each number (or variable).
2. Take the largest power for each prime that appears.
3. Multiply all of the expressions you found in step 2 together.

And here's a video by eHowEducation showing you a different way to find GCF and LCM using the ladder method. The steps are:

1. Write the numbers on one line and make an "L" shape to contain them.
2. Find the smallest prime number that goes into all your numbers and write it next to your numbers.
3. Divide your numbers by the prime and write them below your original numbers.
4. Repeat steps 2 and 3 until there are no common primes that can be removed from the numbers.
5. To find LCM (least common multiply) multiply all the numbers going down on the side (the primes you divided by) and the finals numbers that you can't reduce any more.
6. To find GCF (greatest common factor) multiply all the numbers going down on the side (the primes you divided by).

Think about what we mean when we say the least common multiple (LCM) of numbers. To find the least common multiple of two numbers, we prime factorize both numbers. Then take the higher power of each prime factor that appears in either number, and multiply those together.

If you want more review of this particular concept before moving on, see this explanation on LCMs of numbers.

Now we can extend this concept to polynomials, for which the important thing to remember is that the polynomial is fully factored, because otherwise you may have extra powers of a factor or a factor that shouldn't be there. This isn't a problem if you're comparing monomials, but it can be problematic otherwise.

Speaking of which, let's work on finding the LCM of monomials. The monomials may still have a coefficient, and we treat that just as before. But if we have powers of a variable, we just take the higher power of that variable between all the monomials, and multiply our new coefficient with that. In other words, we want to pretend that the variables are prime numbers and they have their own "slot" in the factorization.

This will be more clear with an example. Let's find the LCM of these two monomials:

$$6a^3b^2$$ $$9ab^5$$

So let's first explicitly break the expressions apart into a coefficient, an $a,$ and a $b:$

$$6 \times a^3 \times b^2$$ $$9 \times a \times b^5$$

Now write out the prime factorizations of the coefficients:

$$2 \times 3 \times a^3 \times b^2$$ $$3^2 \times a \times b^5$$

We start building our "final" LCM by finding the LCM of just the coefficients. The largest power of $2$ is in the first expression (a power of $1$), and the largest power of $3$ is in the second expression (a power of $2$). So, multiply these powers together to see that the coefficient of the LCM will be $2 \times 3^2 = 18.$

Now, look at each variable and take the higher power.

$$6 \times a^3 \times b^2$$ $$9 \times a \times b^5$$

The higher power of $a$ is $3,$ and the second power of $b$ is $5.$ Multiply these together with the $18$ to get a final LCM of $18a^3b^5.$

Let's try another example.

Now things get a little bit trickier. If the expressions we want to find the LCM of are not monomials, we need to take a bit more care.

Maybe, just maybe, one of the factors is a binomial.

So let's try finding the LCM of

$$32x^3(y + 1)$$ $$3x^2(y^2 + 2y + 1)$$

Now what? Well, the coefficients and the powers of $x$ are familiar territory, so let's pull those apart first:

$$2^5 \times x^3 \times (y + 1)$$ $$3 \times x^2 \times (y^2 + 2y + 1)$$

We can find the coefficient of the LCM just like before: the largest power of $2$ is a $5,$ and the largest power of $3$ is a $1,$ so the coefficient is $32 \times 3 = 96.$ Similarly, the largest power of $x$ is a $3,$ so the LCM also has a factor of $x^3.$

Now we need to find what function of $y$ needs to be tossed into the LCM. We can't quite just multiply $y + 1$ and $y^2 + 2y + 1$ together: we have to realize these are different powers of the same binomial! Namely,

$$y + 1 = (y + 1)^1.$$ $$y^2 + 2y + 1 = (y + 1)^2.$$

They're both powers of $y + 1!$ So we just take the larger one, and include a $(y + 1)^2$ factor in our final answer.

Thus, our final LCM is $96x^3(y + 1)^2.$ How about you try one more?