# Thread: Finding Least Common Multiple

1. ## Finding Least Common Multiple

In this problem, I am supposed to find the GCF and LCM of the following:

$\displaystyle 6y^3 + 12y^2z , 6y^2 - 24z^2 , 4y^2 - 4yz - 24z^2$

I determined the GCF to be $\displaystyle 2(y = 2z)$, which the book says is correct. For the LCM however, I am having trouble however. I did the following:

$\displaystyle 6(x^3 + 2y^2z)$ , $\displaystyle 6(y^2 - 4z^2)$, $\displaystyle 4(y^2 - yz - 6z^2)$

$\displaystyle 6(x^3 + 2y^2z) , 6(y + 2z)(y - 2z) , 4(y-3z)(y+2z)$

From this, I determined the LCM to be $\displaystyle 12(x^3 + 2y^2z)(y+2z)(y-2z)(y-3z)$

However, the book says the answer is $\displaystyle 12y^2(y+2z)(y-2z)(y-3z)$

Where does the $\displaystyle 12y^2$ come from?

2. ## Re: Finding Least Common Multiple

After you have factored each term to its irreducible factors, as you probably did to find the Greatest Common Factor, the Least Common Multiple is just that Greatest Common Factor times each of the factors in the original numbers that are NOT part of the Greatest Common Factor.

You say that you found that the GCF to be 2(y+ 2z). (You have "2(y= 2z)" but I presume you meant 2(y+ 2z).)

Now in your fourth line why did you not factor 2(y+ 2z) out of $\displaystyle 6(x^3+ 2y^2z)$?

3. ## Re: Finding Least Common Multiple

Originally Posted by HallsofIvy
After you have factored each term to its irreducible factors, as you probably did to find the Greatest Common Factor, the Least Common Multiple is just that Greatest Common Factor times each of the factors in the original numbers that are NOT part of the Greatest Common Factor.

You say that you found that the GCF to be 2(y+ 2z). (You have "2(y= 2z)" but I presume you meant 2(y+ 2z).)

Now in your fourth line why did you not factor 2(y+ 2z) out of $\displaystyle 6(x^3+ 2y^2z)$?
Hello, yes that was a typo, I meant $\displaystyle 2(y+2z)$

I am not quite sure how to fact $\displaystyle 2(y+2z)$ out of $\displaystyle 6(x^3+2y^2z)$.

4. ## Re: Finding Least Common Multiple

Originally Posted by EngineMan
Hello, yes that was a typo, I meant $\displaystyle 2(y+2z)$

I am not quite sure how to fact $\displaystyle 2(y+2z)$ out of $\displaystyle 6(x^3+2y^2z)$.
You seem to have more than one typo.

In the FIRST line of your post, it is $6(y^3 + 2y^2z)$. In later lines, it becomes $6(x^3 + 2y^2z)$.

If it truly is the former, then $6(y^3 + 2y^2z) = 2(y + 2z)(3y^2)$.

5. ## Re: Finding Least Common Multiple

Originally Posted by Deveno
You seem to have more than one typo.

In the FIRST line of your post, it is $6(y^3 + 2y^2z)$. In later lines, it becomes $6(x^3 + 2y^2z)$.

If it truly is the former, then $6(y^3 + 2y^2z) = 2(y + 2z)(3y^2)$.
AHHH, cripes, now I feel stupid. Yes, that was another typo on my part. This is the bane part of algebra for me, mixing x's and y's, pluses and minuses, etc...up. Also coming up with common multiples for numbers that are too high to be the LCM.

6. ## Re: Finding Least Common Multiple

Originally Posted by EngineMan
AHHH, cripes, now I feel stupid. Yes, that was another typo on my part. This is the bane part of algebra for me, mixing x's and y's, pluses and minuses, etc...up. Also coming up with common multiples for numbers that are too high to be the LCM.
It's not a problem peculiar to algebra...for example many people (myself included) often misspell "friend" F-R-E-I-N-D, and there is this amusing lyric from "Pretty Fly (for a White Guy)":

Well he asked for a thirteen, but they drew a thirty-one...

In other words, people mix up "symbol order" a LOT. It takes time to "wire in the connections" between "what stands for which". Of course, in algebra (or arithmetic, or in guidance systems for nuclear weapons) such mix-ups may have disastrous results.

Here is one thing you can do, to hopefully not be so intimidated by the x's and y's and other strange things:

Recall that 1,2,3,4.... themselves are just SYMBOLS, not really "the actual numbers themselves". The Romans had different symbols, so did the Babylonians, so do the Chinese. So you've already mastered "one level" of letting a squiggly line take the place of an IDEA in your head. Now we're just going to "level 2" of letting a DIFFERENT squiggly line (usually "x") stand for a different idea: a number whose address is unknown. This is like saying "Joe's house" instead of "1431 Bradford Street". It clearly wouldn't do to keep referring to "Joe's house" as "Alice's house", unless you are trying to confuse people.

So...x is x, and y is y. Give them personalities, keep them focused in your mind as DISTINCT entities. And be vigilant: things can "look the same", but not BE the same:

$x^2 + y$ looks a lot like $x + y^2$, but they are very different. The differences can be subtle, but try not to let the tricksey hobbitses fool you.

7. ## Re: Finding Least Common Multiple

Originally Posted by Deveno
It's not a problem peculiar to algebra...for example many people (myself included) often misspell "friend" F-R-E-I-N-D, and there is this amusing lyric from "Pretty Fly (for a White Guy)":

Well he asked for a thirteen, but they drew a thirty-one...

In other words, people mix up "symbol order" a LOT. It takes time to "wire in the connections" between "what stands for which". Of course, in algebra (or arithmetic, or in guidance systems for nuclear weapons) such mix-ups may have disastrous results.

Here is one thing you can do, to hopefully not be so intimidated by the x's and y's and other strange things:

Recall that 1,2,3,4.... themselves are just SYMBOLS, not really "the actual numbers themselves". The Romans had different symbols, so did the Babylonians, so do the Chinese. So you've already mastered "one level" of letting a squiggly line take the place of an IDEA in your head. Now we're just going to "level 2" of letting a DIFFERENT squiggly line (usually "x") stand for a different idea: a number whose address is unknown. This is like saying "Joe's house" instead of "1431 Bradford Street". It clearly wouldn't do to keep referring to "Joe's house" as "Alice's house", unless you are trying to confuse people.

So...x is x, and y is y. Give them personalities, keep them focused in your mind as DISTINCT entities. And be vigilant: things can "look the same", but not BE the same:

$x^2 + y$ looks a lot like $x + y^2$, but they are very different. The differences can be subtle, but try not to let the tricksey hobbitses fool you.
That is interesting, now me, I am usually very good with spelling, but occasionally mix up things when doing algebra. Do the Chinese still use different symbols, or do they use the Arabic numerals that the English-speaking world (and much of the non-English-speaking world) uses today?

8. ## Re: Finding Least Common Multiple

So now I've got a new one that I am stuck on:

$\displaystyle (x/2x^2 + 3xy + y^2) - (x - y / y^2 - 4x^2) + (y / 2x^2 + xy - y^2)$

To find the Lowest Common Denominator, or the Lowest Common Multiple of the denominators of the fractions, I factored each denominator:

$\displaystyle 2x^2 + 3xy +y^2 = (2x + y)(x + y)$
$\displaystyle y^2 - 4x^2 = (y + 2x)(y - 2x)$
$\displaystyle 2x^2 + xy - y^2 = (2x - y)(x + y)$

At first I had determined the LCD to be $\displaystyle (2x + y)(2x - y)(y - 2x)(x + y)$. I went through a whole ton of trying to work out the problem, but then looked at the answer, and it has in the denominator just $\displaystyle (2x + y)(2x - y)(x +1)$. Is $\displaystyle (y - 2x)$ left out because it is the same as $\displaystyle (2x - y)$ but just multiplied by $\displaystyle -1$?

9. ## Re: Finding Least Common Multiple

Not sure if this has already been pointed out, but if you already have the GCF of the pair, then you get the LCM by the formula:

GCF(a, b) x LCM (a, b) = a x b

That is, multiply the entities together and divide by the GCF, and you're there.

Oh, the term "lowest common denominator" is probably best reserved for calculation of the sum of numerical fractions because the "denominator" is the bottom of the sum of the fractions being calculated. In the context of polynomials, or indeed anywhere where you are not actually calculation sums of fractions, the term "lowest common multiple" is far more explicit of what you are trying to do.

The problem, I appreciate, is that "lowest common denominator" is one of the techniques that is taught by rote early in one's mathematical training, and so the phrase has its own path burned deeply into the neurons, and thus there is the danger of it being trotted out in situations where its use is not strictly applicable.

10. ## Re: Finding Least Common Multiple

Originally Posted by EngineMan
So now I've got a new one that I am stuck on:

$\displaystyle (x/2x^2 + 3xy + y^2) - (x - y / y^2 - 4x^2) + (y / 2x^2 + xy - y^2)$

To find the Lowest Common Denominator, or the Lowest Common Multiple of the denominators of the fractions, I factored each denominator:

$\displaystyle 2x^2 + 3xy +y^2 = (2x + y)(x + y)$
$\displaystyle y^2 - 4x^2 = (y + 2x)(y - 2x)$
$\displaystyle 2x^2 + xy - y^2 = (2x - y)(x + y)$

At first I had determined the LCD to be $\displaystyle (2x + y)(2x - y)(y - 2x)(x + y)$. I went through a whole ton of trying to work out the problem, but then looked at the answer, and it has in the denominator just $\displaystyle (2x + y)(2x - y)(x +1)$. Is $\displaystyle (y - 2x)$ left out because it is the same as $\displaystyle (2x - y)$ but just multiplied by $\displaystyle -1$?
"Is $\displaystyle (y - 2x)$ left out because it is the same as $\displaystyle (2x - y)$ but just multiplied by $\displaystyle -1$?"

Yes that would be the case.

One final point: is $\displaystyle (x + 1)$ a typo for $\displaystyle (x + y)$ in the answer supplied by the book?

11. ## Re: Finding Least Common Multiple

Originally Posted by Matt Westwood
Not sure if this has already been pointed out, but if you already have the GCF of the pair, then you get the LCM by the formula:

GCF(a, b) x LCM (a, b) = a x b

That is, multiply the entities together and divide by the GCF, and you're there.

Oh, the term "lowest common denominator" is probably best reserved for calculation of the sum of numerical fractions because the "denominator" is the bottom of the sum of the fractions being calculated. In the context of polynomials, or indeed anywhere where you are not actually calculation sums of fractions, the term "lowest common multiple" is far more explicit of what you are trying to do.

The problem, I appreciate, is that "lowest common denominator" is one of the techniques that is taught by rote early in one's mathematical training, and so the phrase has its own path burned deeply into the neurons, and thus there is the danger of it being trotted out in situations where its use is not strictly applicable.
Sorry, ignore all this, apologies, you *are* trying to add fractions! D'oh. Apologies, as I say.

12. ## Re: Finding Least Common Multiple

Originally Posted by Matt Westwood
"Is $\displaystyle (y - 2x)$ left out because it is the same as $\displaystyle (2x - y)$ but just multiplied by $\displaystyle -1$?"

Yes that would be the case.
Coolbeans; that is a bit confusing though.

One final point: is $\displaystyle (x + 1)$ a typo for $\displaystyle (x + y)$ in the answer supplied by the book?
:::BANGS HEAD::: It's stuff like this that is why I end up making mistakes in these problems. Yes, the answer is $\displaystyle (x + y)$ not $\displaystyle (x + 1)$

$\displaystyle (3x^2 + xy)$
---------------------------
$\displaystyle (2x + y)(2x - y)(x + y)$

However, I keep getting:

$\displaystyle x^2 + xy + y^2$
-----------------------------
$\displaystyle (2x + y)(2x - y)(x + y)$

I got this from doing the following:

$\displaystyle x(2x - y) - (x - y)(x + y) + y(2x + y)$
------------------------------------------------
$\displaystyle (2x + y)(2x - y)(x + y)$

This is from the factoring:

$\displaystyle x / (2x^2 + 3xy + y^2) = x / (2x + y)(x + y)$
$\displaystyle (x - y) / (y^2 - 4x^2) = (x - y) / (y + 2x)(y - 2x)$
$\displaystyle y / (2x^2 + xy - y^2) = y / (2x - y)(x + y)$

So I multiply x by (2x - y), (x - y) by (x + y), and y by (2x + y).

This gives me $\displaystyle (2x^2 - xy) - (x^2 - y^2) + (2xy + y^2)$ which leads to $\displaystyle x^2 + xy + 2y^2$ in the numerator.