Least Common Multiple (LCM)

Yesterday, we covered the concept of a multiple. Today, we’re going to look at finding the least common multiple (LCM) of two numbers.

Since we created a list of multiples for 4 yesterday, let’s start with that list.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

Now let’s find the LCD for 4 and 5. We’ll need to make a list of some of 5’s multiples.

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

Like we did when we were trying to find the GCF for two numbers, we going to stack our two lists together.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
5: 5, 10, 15, 20, 25, 30, 35, 40, 45,  50

Now, let’s highlight the numbers on both lists.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

Both 20 and 40 are highlighted in the lists, but because 20 is smaller, it is the Least Common Multiple for 4 and 5.

Being able to determine LCM will help you as you work with fractions. It will allow you to create equivalent fractions quickly and well.

What is a multiple?

Let’s start out today by defining “multiple”.

Multiple: the result of multiplying the target number by another number

It sounds simple enough, right? As simple as the definition is, it’s amazing how many people confuse “multiple” and “factor“. A little care in your work will help you avoid making this common mistake.

To find multiples of a number, start multiplying it by other numbers. For example, let’s look at some of the multiples for 4:

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

If you’re looking at this list and thinking this list looks a lot like the multplication facts for 4, then you’re right.

Multiples are really that simple.

Prime factorization

The first post in this mini-series addressed factoring using multiplication facts and the rules of divisibility. Today, we’re going to factor a number into its primes.

A prime number is any number that can be divided only by itself and 1. Some people find this a more useful way to find common denominators,and it can actually make simplifying radical expressions much simpler.To start factoring a number to its primes, we need to either apply the multiplication facts or the rules of divisibility to it.

Let’s use 72 for this example. 72 is 8 * 9.

72
/\
8 9

Neither 8 nor 9 is prime, so we’re going to factor both of them. We know that 8 is 4 * 2; and 9 is 3 * 3, so let’s add those to our factor tree.

72
/ \
8   9
/\  /\
2 4  3 3

Now we’re getting somewhere! Both 2 and 3 are prime, leaving us only the 4 to factor. The factor tree looks like this now.

72
/ \
8   9
/ \  /\
2 4  3 3
/\
2 2

I’ve bolded the prime numbers at the end of each branch so we can see them clearly. The prime factorization of 72 is 2 * 2 * 2 * 3 * 3.

If you use this method to find GCFs, then you’ll need to find all the primes both numbers have in common and multiply them back together. For example, if you were comparing 72 to 12, you’d find that both numbers have two 2s and a 3 in their list of prime factors. 2 * 2 * 3 equals 12, so 12 would be the GCF for 12 and 72.

Greatest Common Factor (GCF)

Yesterday, we reviewed how to factor a number. Today, we’re going to use that skill to work on determining the greatest common factor (GCF) for two numbers. This skill is most often useful when trying to come up with a common denominator for two fractions that you want to add or subtract.

Let’s find the GCF for 12 and 15.

Yesterday, we factored out 12. Let’s review that list.

12: 1, 2, 3, 4, 6, 12

Now, let’s factor 15. We have the identity factors of 1 and 15. A quick run through the rules of divisibility tells us that 15 is divisible by 3 and 5. Let’s look at our factor list for 15.

15: 1, 3, 5, 15

To determine the greatest common factor for 12 and 15, we stack the two lists.

12: 1, 2, 3, 4, 6, 12

15: 1, 3, 5, 15

Now we determine what numbers are on both lists.

12: 1, 2, 3, 4, 6, 12

15: 1, 3, 5, 15

Both 1 and 3 are bolded on both lists, but becasue 3 is the larger number, it is the greatest common factor of 12 and 15.

Factoring for the masses

I work primarily with students who are studying anything from pre-algebra to geometry, so I spend a lot of time reminding them how to factor. It’s amazing how often this skill is required. You need it to work with fractions, polynomials, and radical expressions.

First, let’s start with a definition. What is a factor?

Factor: a number that can be multiplied by another number to equal the target number

This is where it’s helpful to be fluent in your multiplcation facts and to be comfortable with the rules of divisibility.

To determine the factors for a number, it’s helpful to write them all out in a line. Let’s practice by factoring 12.

12:

We can practice our rules of divisibility while we’re at it. Does 12 pass the rule for divisibility by 2? It ends in a 2, so it does. It’s 2 * 6. Let’s add that to the list.

12: 2, 6

Next, let’s check if it’s divisible by 3. 1 + 2 equals 3, so 12 is divisible by 3. It’s 3 * 4. The list now looks like this.

12: 2, 6, 3, 4

3 and 4 are only one digit apart. We have almost completely exhausted our list of factors. We just need to add the identity factors of 1 and 12 to make our list complete.

12: 2, 6, 3, 4, 1, 12

Factor lists are easiest to use if they’re in order, so let’s wrap up this factoring session by putting the factors in order from least to greatest.

12: 1, 2, 3, 4, 6, 12

Once you can factor numbers, finding a greatest common factor and factoring polynomials becomes a lot more simple.

Rules of divisibility

It can sometimes be useful to be able to quickly figure out what numbers a given number can be divided by. For example, you may be trying to simplify a fraction or a radical expression. In these situations, it’s a good idea to remember the rules of divisibility.

Rules of divisibility

• 2: The ones digit is 2, 4, 6, 8, or 0.
• 3: Add the digits in the number together. If the resulting number is divisible by 3, then the original number is divisible by 3.
• 4: If the last two digits in the number are divisible by 4, then the original number is divisible by 4.
• 5: The ones digit is 5 or 0.
• 6: If the number is divisible by both 2 and 3, then it is divisible by 6.
• 8: If the last three digits are divisible by 8, then the number is divisible by 8.
• 9: Add the digits in the number together. If the resulting number is divisible by 9, then the original number is divisible by 9.
• 10: The ones digit is 0.

There have been several recent attempts to create a simple rule for checking for divisibility by 7, but so far nothing has proven simple enough to include here.

To check if a number is divisible by one of the numbers on this list, simply run it through the test. For example,  if you want to know if 115 is divisible by 3, add the digits together. In this case, 1 + 1 + 5 = 7. 7 is not divisible by 3, so 115 is not divisible by 3.

Try taking a number and testing it for each rule.  I’ll demonstrate using 249. Let’s see what it can be easily divided by.

• 2: The last digit of 249 is 9. 9 is not 2, 4, 6, 8, or 0, so 249 is not divisible by 2.
• 3: 2 + 4 + 9 = 15. 15 is divisible by 3, so 249 is divisible by 3.
• 4: 49 is not evenly divisible by 4, so 249 is not divisible by 4.
• 5: The last digit of 249 is 9. 9 is not 5 or 0, so 249 is not divisible by 5.
• 6: We have already proven that 249 is not divisible by 2, so it is also not divisible by 6.
• 8: 249 divided by 8 gives a remainder, so it is not divisible by 8.
• 9: 2 + 4 + 9 = 15. 15 is not divisible by 9, therefore 249 is not divisible by 9.
• 10: The last digit of 249 is 9. 9 is not 0, so 249 is not divisible by 10.

According to our work, 249 is divisible only by 3.

Once you get the hang of it, checking the rules of divisibility goes pretty quickly.

The 30-60-90 triangle

A couple of weeks ago, we looked at the ratio for solving problems involving triangles where the angles measured 45°, 45°, and 90°.

Today, I’d like to look at the ratio for the sides of a triangle where the angles measure 30°, 60°, and 90°.

Perhaps it might be better to talk about the 30-90-60 triangle. I realize I’ve set them in a different order, but you’ll see why in a moment.

The ratio for the 30-90-60 triangle is 1: 2: √3. That is, 1 represents the short leg of the triangle. The 2 represents the hypotenuse. The √3 represents the long leg of the triangle. Now you see why I switched them around.

Like the 45-45-90 triangle, you put the measure of the side you already know, and place a variable over the side you need to know. Then you solve it like a proportion, just as you did with the 45-45-90 triangle.

The number line as a means of ordering and comparing numbers

So far, we’ve covered the number line’s skill at telling us the location of an object and the distance between two objects. Now, let’s take a look at using the number line to sort numbers from least to greatest.

Say we want to organize these bunnies so the numbers on their shirts go in order from least to greatest.

 Since 13 is larger than 3, we set up the bunnies in the following row:

 We want to check our order, so we set the bunnies on the number line.

Oops! The bunny at the end of the line should actually be at the front of the line. What happened?On the number line, numbers on the left have a lower value than number on the right. As a result, positive numbers are greater than negative numbers. If both have a negative sign, then the apparent smaller number actually has the greater value.

2 is smaller than 9 when both are positive, but because -2 is to the right of -9, it is actually the larger.

This concludes Dead Bunny’s series on number lines. Once I get some issues straightened out, these posts should come together to form Dead Bunny’s first article.

Solving proportions

There are certain skills that you absolutely need to survive in math: order of operations, the distributive property, the ability to solve for a variable, creating equivalent fractions, and solving proportions. When you have a handle on these skills, most math should be within your grasp.

Today, we’re going to focus on the overly useful proportions. Proportions help you solve percent, metric conversions, scale, and unit price problems. The basic appearance of your standard proportion equation looks like this:

To start solving, you cross-multiply and set the products equal to each other.

ad = bc

From here, you divide both sides of the equation by whatever number is sitting beside the variable you’re solving for. For example, if you were solving for a, then the resulting equation would look like: a = (bc)/d.

 

More on fractions

I’m still working on my fraction posts and articles, but I found a set of articles that go along with my equivalent fraction post and should at the very least get you started.

• A part of the whole