| Multiplying | Prime numbers |
| Long Multiplication | Greatest Common Factors (GCF) |
| Dividing | Least Common Multiples (LCM) |
| Long Division | More Practice |
Let’s talk a bit about the concept of multiplication (“times-ing”) and division (“grouping”).
Multiplying
You can think of multiplication like blocks in a box, where you have a number of blocks along the top, a number of blocks down the side, and multiplying is adding up all the blocks in the box.
For example, let’s say we have $ 3$ pairs of pink shoes and $ 3$ pairs of gray shoes. Thus, we have $ 2$ different colors of shoes, and $ 3$ pairs of shoes for each color:
Again, we have $ 2$ groups of shoes with $ 3$ shoes of the same color in each group. Multiplication is quite simple: to get the total number of shoes we have, we multiply the number of groups ($ 2$) by the number of things in each group ($ 3$) to get $ 6$. That’s all multiplication is!
The numbers you multiply are called factors and the resulting number is a product or a multiple.
We can write multiplication in several different ways. We can use an “$ \times $”, a “•”, or an “$ *$” (asterisk). We can also put the numbers we are multiplying in parentheses. All these ways mean the same thing with multiplication. We can write multiplication in so many different ways, since it’s used for so many applications. Here are examples:
| $ 3\times 2=6$ | $ \displaystyle 3\cdot 2=6$ | $ \displaystyle 3*2=6$ | $ \left( 3 \right)\left( 2 \right)=6$ |
This next table is one of the most important tables you’ll ever learn – trust me, if you know this well, it will help you your whole life!
Undoubtedly, in school, you will go over these over and over again, and there’s really no “cool” way to learn them, except to remember that every time we go up in a number in multiplication, we are just adding that many more to what we had (like when we go from “$ 5\times 3$” to “$ 5\times 4$”, we just add $ 5$). Flashcards will help if you want to be ahead of the game!
Memorize your multiplication facts:
The factors are on the outside, and the products or multiples are on the inside. For example, if we start with $ 5$ on the left and look for the $ 4$ on the top (these are both factors), they both point to $ 20$ (which is the product). Thus, $ 5\times 4=20$. Note that it doesn’t matter which number we start with: $ 5\times 4=4\times 5=20$.
Learn these multiplication facts well – you will use them your entire life! Again, it’s a great idea to either use flash cards to learn them, or have someone quiz you.
Here are a couple of multiplication hints:
- Any number multiplied by $ 0$ is $ 0$.
- Any number multiplied by $ 1$ is that same number.
- Any number multiplied by $ 2$ will give you an even number.
- If a number’s digits add to a multiple (product) of $ 3$, $ 3$ goes into that number (even big numbers!). For example, for the number $ 27$, $ 2+7=9$, and $ 3$ goes into $ 9$, so $ 3$ goes into $ 27$.
- If the number formed by the last two digits of a number are divisible by $ 4$, then the whole number is divisible by $ 4$. For example, the number $ 403232$ is visible by $ 4$, since $ 32$ (the last two digits) is divisible by $ 4$. Pretty cool!
- Any number that ends in either $ 0$ or $ 5$ is divisible by $ 5$. This is more obvious.
- If a number is even and the sum of the digits is divisible by $ 3$, that number is divisible by $ 6$. Makes sense, with the rules of three above.
- If the number formed by the last three digits of a number are divisible by $ 8$, then the whole number is divisible by $ 8$. For example, $ 12104$ is divisible by $ 8$, since $ 104$ is divisible by $ 8$.
- If a number’s digits add to a multiple (product) of $ 9$, $ 9$ goes into that number. For example, for the number $ 882$, $ 8+8+2=18$, and $ 9$ goes into $ 18$, so $ 9$ goes into $ 882$.
- If a number ends with a $ 0$, it’s divisible by $ 10$.
- Even numbers multiplied by any other number are always even.
- If you reverse the digits of any number and subtract the two numbers, you get a multiple of $ 9$. For example, $ 85-58=27$, and $ 9$ goes into $ 27$. Weird!
Long Multiplication
Now I’m going to show you how to do “long” multiplication, which is multiplying two or more numbers that are greater than 10. Let’s say your mom is buying $ 48$ place settings that are $ \$29$ each. How much is she spending before tax?
To do “long” multiplication, you have to be very patient, and do a series of steps. It’s a little odd, because first you have to do multiplication steps, and then addition steps, then multiplication, then addition, and so on. We always start on the right, and work towards the left (backwards from reading again). Here we go – understand each step and don’t rush!
One more important thing to remember: always put the longer number (number with more digits) on top. This will make your life much easier! In this case, since both of our numbers have two digits, it didn’t matter which number is on top.
Memorize these rules!
Thus, she is spending $ \$1392$. Now, using this technique, you can multiply any two numbers together. You just reapply what you’ve done; you may have to add more than two rows together to get the final answer, but it’ll work. This is why I love math: I get to keep reapplying what I’ve learned, instead of learning a bunch of new stuff, like I had to do in history, for example.
It’s much easier if you multiply with numbers that end in $ 0$. You can sort of ignore the $ 0$’s at the end of each number, and then add that number of $ 0$’s when you’re done. Let’s multiply $ 160$ by $ 20$:
Dividing
Now let’s talk about division, which is the opposite of multiplication (like subtraction is to addition). When we divide something by something, we are basically dividing into groups. You should know your division facts from your multiplication facts (backwards!)
For example, let’s take the shoe example again. If we have $ 6$ pairs of shoes, and we have $ 2$ different colors of shoes, how many pairs of shoes do we have in each color? See how we will come up with $ 3$ pairs of shoes for each color? This is all dividing is: putting things into groups.
Here are the different ways we show division:
| $ 6\div 2=3$ | $ \displaystyle \frac{6}{2}=3$ | $ 6/2=3$ | $ 2\overset{3}{\overline{\left){6}\right.}}$ |
(For the last one, I remember that we put the number that is first or on top inside the “house”; since it’s “superior”, it gets to be “inside”.)
The $ 6$ the is the dividend, or the number we are dividing “into”, the $ 2$ is the divisor, or the number of “groups”, and the $ 3$ is the quotient, or the answer, like the product is the answer for multiplication. See how a divisor can also be thought of as a factor? Also note that a division problem is written like $ \displaystyle \frac{6}{2}$ is a fraction, which we’ll address in a later section.
Note that $ 6$ divided by $ 2$ is $ 3$, and also $ 6$ by $ 3$ is $ 2$, since $ 2\times 3=6$, and we are “undoing” the multiplication. We can look at the $ 6$ pairs of shoes and see that we have $ 3$ pairs of shoes of each color, so we have $ 2$ different colors. Makes sense!
Know your division facts from your multiplication facts:
The division table is identical to the multiplication table, except you start from the inside and move to the outside (like we did with the subtraction table). For example, if you start with $ 12$ (the dividend), and you want to divide by $ 4$ (on the left – the divisor), you look up to the top and see the answer is $ 3$ (the quotient).
Learn these division facts well; again, you will use them your entire life! Again, it’s a great idea to either use flash cards to learn them, or have someone quiz you. And you’ll find once you learn the multiplication facts, you’ll also know the division facts.
Long Division
Now let’s try some “long” division, just like we did long multiplication above.
Let’s supposed that our parents say that we can spend $ \$275$ on a birthday party at a skating rink, and the skating rink will charge $ \$17$ per person for skates and food. How many people can we invite to the party?
This is a case of long division, since we’re dealing with larger numbers (more specifically, a divisor greater than $ 10$). Again, we can write division may ways:
| “dividing $ 275$ by $ 17$” | $ 275\div 17$ | $ \displaystyle \frac{{275}}{{17}}$ | $ 17\overline{\left){{275}}\right.}$ |
We need to write it the last way to do the long division. Let’s go for it; we’ll actually have to work with division, multiplication, and subtraction to do this problem:
Memorize these rules!
If you can do this, you can do any long division problem! We will work on going further to create decimal answers (instead of having a remainder) in the Decimals section if the numbers don’t go into each other perfectly, like this one.
Like multiplication, dividing is much easier if you have $ 0$’s at the end of the numbers; you can just cross out the $ 0$’s on the top and bottom before doing the division.
Let’s divide $ 75000$ by $ 250$:
Prime numbers, Greatest Common Factors, and Least Common Multiples
At this time, I also want to give you a couple more concepts that you’ll need throughout your math career.
A prime number is a number that is only divisible by $ 1$ and itself: $ 2,3,5,7,11,13,17,$ and so on. Notice that no even numbers can be prime, except for $ 2$. Every “regular” number (like $ 1,2,3,4$ and so on) can be factored into prime numbers. Prime numbers have exactly two factors: $ 1$ and itself.
A composite number is a number that is not a prime number. Composite numbers are $ 4,6,8,9,10,12,$ and so on.
Note that the number “ $ 1$” fits into its own category, and is neither prime nor composite. It is not prime, since it technically doesn’t have two factors.
Greatest Common Factor (GCF)
The Greatest Common Factor (or GCF) of numbers is just what it says: it’s the greatest number (other than $ 1$) that goes into the numbers without any remainders. It can be one of the numbers, for example, the greatest common factor of $ 3$ and $ 6$ is $ 3$, since $ 3$ is the largest number that goes into both $ 3$ and $ 6$ perfectly. The Greatest Common Factor is sometimes called the Greatest Common Divisor.
To get the GCF of numbers, you can list all the prime factors of the numbers, match up factors on both sides, and multiply these to get the greatest factor.
Let’s find the GCF of $ 12$ and $ 18$. “Drill down” by starting with any two factors; you can always try to start with $ 2$ for even numbers. The order doesn’t make a difference; divide each number down until there are only prime numbers.
I like to create prime factor trees, where we “dissect” the numbers and “branch out” to get down to the prime factors. The lowest factors (the ones that can’t be divided by any number except for itself and $ 1$) are the lowest “leaves” on the tree and are the prime factors:
To get the GCF, see what factors are on all numbers (factorizations), circle them, and then multiply them together (just on one side). Since we have a “$ 2\times 3$” on both sides, the GCF is $ 6$ ($ 2\times 3$). Note that we don’t circle the other $ 2$ under the $ 12$, and other $ 3$ under the $ 18$, since we don’t have matches for them on the other side. But we can circle two of the same numbers if they match up on both sides.
Thus, $ 6$ is the largest factor of (or number that goes into) $ 12$ and $ 18$. We’ll use the GCF later when we want to reduce a fraction to its “simplest” form.
Here is another example:
Find the GCF of $ 20$, $ 28$, and $ 56$:
Create a factor tree and circle any matches of prime factors that are in all three trees. Then multiply these together to get the GCF:
To get the GCF, we see what factors are under all numbers (factorizations), circle them, and then multiply them together (just on one side). Since we have a “$2\times 2$” circled under all numbers, the GCF is $ 4$ ($2\times 2$). Note that we could circle two of the same numbers (the $ 2$’s) since they were part of the factorization under all the numbers.
Least Common Multiple (LCM)
The Least Common Multiple (or LCM) is used more often, since we’ll see later that we’ll use it to add and subtract fractions. The Least Common Multiple is also called the Least Common Denominator (LCD), when used with fractions.
Let’s find the LCM of $ 4$ and $ 6$. To do this, find the smallest number that they both go into (multiples). Begin by writing down all the multiples, starting with the actual number:
MULTIPLES of $ \boldsymbol{4}$: $ \displaystyle 4,8,\boldsymbol{{12}},16,20,\boldsymbol{{24}},28,32\ldots $
MULTIPLES of $ \boldsymbol{6}$: $ \displaystyle 6,\boldsymbol{{12}},18,\boldsymbol{{24}},30\ldots $
Then find the lowest number that is in both lists. Note that the smallest multiple of both the numbers is $ 12$; another common multiple is $ 24$, but this not the smallest. You can always get a common multiple of numbers (you can have more than two!) by multiplying them together, but this is not always the smallest one.
We can also use a factor tree to find the LCM. For each prime factor, find where it occurs the most often under a number, even if it’s only once, and circle these factors. Note that if we have circled one or more prime factors, we don’t circle it again under any other number. Then multiply them all across to get the LCM. Here’s the example with $ 4$ and $ 6$:
Then multiply all the factors that we’ve circled to get $ 2\times 2\times \color{#0070C0}{3}\,\,\color{black}{{=12}}$. Thus, the LCM of $ 4$ and $ 6$ is $ 12$.
Now find the LCM of $ 14$, $ 18$, and $ 168$ by using factor trees.
Again, list the prime factors across all numbers, and for each prime factor, circle them only if they occur the most often under that number, and don’t repeat circling factors across numbers:
Make sure all the prime factors are covered ($ 7$, $ 2$, and $ 3$); it looks like they are. Also note that we only circled the $ 7$ under one of the numbers ($ 14$), since we don’t repeat circled numbers across different numbers.
Then multiply everything that’s circled to get $ \color{#7030A0}{7}\times \color{#0070C0}{3}\times \color{#0070C0}{3}\times \color{#804040}{2}\times \color{#804040}{2}\times \color{#804040}{2}\,\,\color{black}{{=\,504}}$. Thus, the LCM of $ 14$, $ 18$, and $ 168$ is $ 504$.
Learn these rules and practice, practice, practice!
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On to Decimals – you are ready!!