Factoring: How to Be the Hero of Pizza Night

Lesson Plan - Get It!

Audio:

What Is a Factor?

A factor is any number that divides evenly into another number.

We can also say that numbers are divisible -- able to be divided evenly -- by their factors.

For example, 2 is a factor of 10 because 10 divided by 2 is the whole number 5. On the other hand, 2 is not a factor of 11 because 11 divided by 2 is 5.5, which is not a whole number.

In other words, 10 is divisible by 2, but 11 is not divisible by 2.

When you hear the word factor, you need to think divide.

To make Family Pizza Night fair, Sarah needs to cut enough slices of pizza so that everyone can have the same number of slices. If she succeeds, the number of slices and the number of people will both be factors of the number of total pizza slices.

Shortcuts

Factoring sometimes involves a lot of guesswork, especially when you are just starting, because you need to divide the two numbers before you can say for sure that one is indeed a factor of the other.

As you factor more, you will start to recognize several shortcuts.

So, if you have several hungry people waiting for you to cut the pizza so they can eat, you will find the following ways to easily identify some specific patterns and factor quickly helpful!

The Number 2

A number of any size is divisible by 2 if the last digit is even (0, 2, 4, 6, and 8). If the last digit is even, then the whole number is even.

In fact, by definition, an even number is divisible by 2.

Examples:

• 14
• 72
• 148
• 1,006
• 1,000,000

The Number 3

A number is divisible by 3 if the digits add up to a number that is divisible by 3 (3, 6, 9, 12, 15, etc.). This is a pretty neat trick!

For some numbers, this is much faster than dividing because you can probably add up the digits quickly in your head.

Examples:

891 IS divisible by 3 because the digits add up to be 18, which is divisible by 3.

1,000 IS NOT divisible by 3 because the digits add up to be 1, which is not divisible by 3.

The Number 4

You can tell if a number is divisible by 4 by annexing the last 2 digits of the number, the tens place and the ones place.

This works because any number that ends in at least 2 zeroes (like 100 or 10,000) is divisible by 4, so just subtracting the last 2 digits does not change the divisibility of the number. For large numbers, this is much quicker because you only have to divide a 2-digit number.

Of course, this only applies to even numbers. An odd number is never a factor of an even number.

• As a challenge, can you figure out why this is?

Examples: 

342 IS NOT divisible by 4 because 42 divided by 4 is 10.5, which is not a whole number.

343 IS NOT divisible by 4 because it is odd.

344 IS divisible by 4 because 44 divided by 4 is 11, which is a whole number.

The Number 5

If know how to count by 5's, you know that there is a pattern: 5, 10, 15, 20, 25, 30, 35, 40...

• Do you see it?

Every factor alternately ends with a 5 or a 0. This is our shortcut for 5. Only numbers that end in 5 or 0 are divisible by 5.

Examples:

• 75
• 810
• 9,745
• 10,000

The Number 6

A number is divisible by 6 if it is divisible by 2 AND 3. It has to be even as well as have a digit sum divisible by 3.

Examples:

180 IS divisible by 6 because it is even, and the sum of the digits is 9, which is divisible by 3.

182 is even, but the sum of the digits is 11, which is not divisible by 3, so it IS NOT divisible by 6.

183 is divisible by 3 since the sum of the digits is 12, but it is not even, so it IS NOT divisible by 6.

• Are these making sense to you?

There are a lot to remember, but knowing these shortcuts will save you SO MUCH time in finding factors. Only 3 more!

There is no special trick to 7. You have to do it the long way.

The number 8 has an alternate method, but it is not really a shortcut.

• As a challenge, can you figure out another way to see if 8 is a factor other than dividing?

The Number 9

The number 9 works exactly the same as 3. The only difference is that the digits have to add up to a number divisible by 9 (9, 18, 27, 36, 45, etc.).

Examples:

999 IS divisible by 9 because the digits add up to 27, which is divisible by 9.

939 IS NOT divisible by 9 because the digits add up to 21, which is not divisible by 9.

The Number 10

The number 10 is probably the easiest shortcut because only numbers that end in a 0 are divisible by 10.

Examples:

• 90
• 200
• 9,730
• 100,000

There are many other alternative methods for larger numbers like 11, 60, and 100.

• Can you figure them out?

Last Notes

There are a few important factor facts that you may have even noticed yourself!

1. A factor is always smaller than the number for which it is a factor.

Though you can divide a larger number into a smaller number, you will get a decimal number as the quotient. A decimal is not technically divisible since it is not a whole number.

2. The number of slices Sarah needs to cut is a multiple of the number of people who are going to eat.

A multiple is different from a factor, but they are often confused.

Every number has a limited amount of factors because there is only a certain quantity of numbers that divides evenly. There is an infinite number of multiples for each number because you can multiply by any whole number to get a multiple.

However, factors and multiples are related, which is probably why they are sometimes mixed up. In fact, the numbers in a multiplication problem are called, wait for it...FACTORS!

When you are ready, keep going in the Got It? section.