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Math Lecture Notes
Arithmetic Skills

Divisibility Rules link to here link to index

Some divisibility rules you already know:

Other divisibility rules may be new to you:

Let's apply the divisibility rules to 13,512.

Example 1

Is 13,512 divisible by... 2?       3?       4?       5?       6?       9?      10?

Divisible by 2? Yes, because the one's place value digit is 2, which is even.

Divisible by 3? Yes, because the sum of digits is 1 + 3 + 5 + 1 + 2 = 12 and three goes into 12.

Divisible by 4? Yes, because two-digit number formed from its ten's and one's digits is 12 and four goes into 12.

Divisible by 5? No, because the one's place value digit is not zero or five.

Divisible by 6? Yes, because it was divisible by both 2 and 3.

Divisible by 9? No, because the sum of digits is 1 + 3 + 5 + 1 + 2 = 12 and nine does not go into 12.

Next let's apply the divisibility rules to 2,016.

Example 2

Is 2,016 divisible by... 2?       3?       4?       5?       6?       9?      10?

videos

Kahn Academy

Divisibility Tests for 2, 3, 4, 5, 6, 9, 10

Bittinger Chapter Tests, 11th Edition

Chapter 2 Test, Problem 6: Determine whether 1,784 is divisible by 8.

Chapter 2 Test, Problem 7: Determine whether 784 is divisible by 9.

Chapter 2 Test, Problem 8: Determine whether 5,552 is divisible by 5.

Chapter 2 Test, Problem 9: Determine whether 2,322 is divisible by 6.

The rule for six works because 2 × 3 = 6.

There are yet more divisibility rules, but these are the ones that are easy to use.

Finding Factors link to here link to index

There are two tasks involving factor finding. In some situations we want to find all the factors. In other situations we want to find the prime factorization.

(In this class the most common use for finding all the factors is considering how to reduce a fraction, and the most common use for finding the prime factorization is to create a common denominator for fractions.)

Finding All the Factors

We find all the factors of a number by making a two-column list. Count up in the first column. List any matching factors in the second column. When the columns get to the same value we can stop.

Example 3

Find all the factors of 66.

Our first column counts up from 1 to 10.

166
233
322
4not a factor
5not a factor
611
7not a factor
8not a factor
9not a factor
10not a factor

The next value for the first column would be 11, which is already listed in the second column. So we can stop.

Example 4

Find all the factors of 24.

videos

Kahn Academy

Finding Factors of a Number

Bittinger Chapter Tests, 11th Edition

Chapter 2 Test, Problem 1: Find all the factors of 300.

Finding the Prime Factorization

We find all the prime factorization of a number by making a factor tree and noting the "leaves".

Factor Trees

Remember factor trees?

Here is one factor tree for 48.

rounding example

Example 5

Try writing a different factor tree for 48 that does not start with 6 and 8.

Prime Factorization

Either circle or "bring down" the leaves of your factor tree so you do not make a careless mistake and forget any of them when writing your answer.

To be polite, list the prime factors in order. Write them as a product (separated by × symbols).

Optionally, you may show off your fluency with exponents by writing the prime factors as compactly as possible using exponents.

Example 6

Find the prime factorization of 48.

Looking at the factor trees for 48 we see the prime factorization is 2 × 2 × 2 × 2 × 3 = 48

This can also be written as 24 × 3 = 48

Example 7

Find the prime factorization of 66.

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Chapter 2 Test, Problem 4: Find the prime factorization of 18.

Chapter 2 Test, Problem 5: Find the prime factorization of 60.

Decimal Point Scoots link to here link to index

When we multiply or divide by powers of ten it only scoots the decimal point.

Example 8

Multiply 123.456 by 10

Multiply 123.456 by 100

Multiply 123.456 by 1,000

Example 9

Divide 123.456 by 10

Divide 123.456 by 100

Divide 123.456 by 1,000

When we multiply or divide by powers of one-tenth it also scoots the decimal point, but in the other direction.

Example 10

Multiply 123.456 by 0.1

Multiply 123.456 by 0.01

Multiply 123.456 by 0.001

Example 11

Divide 123.456 by 0.1

Divide 123.456 by 0.01

Divide 123.456 by 0.001

videos

Kahn Academy

Multiplying a Decimal by a Power of 10

Dividing a Decimal by a Power of 10

Dividing a Decimal by a Power of 10: Pattern

Multiplying Decimals by 10, 100, and 1000 (worksheet)

Dividing Decimals by 10, 100, and 1000 (worksheet)

One-Step Equations link to here link to index

Solving One-Step Equations

In Math 20 the only equations we solve are one-step equations in which the letter is either multiplied or divided by a number (to equal another number).

To solve these, "undo" what is attached to the letter by doing the opposite.

Example 12

Solve 5 × y = 35

We "undo" a multiply-by-5 with a divide-by-5. To be fair, we treat both sides of the equation the same.

Example 13

Solve 4 × b = 20

solving multiplication example

How does this picture help us solve for b? Where is division for both sides hiding?

Example 14

Solve p ÷ 6 = 8

solving division example

How does this picture help us solve for b? Where is multiplication for both sides hiding?

Horizontal and Vertical Format

Let's solve for y with the horizontal format.

Example 15

Solve 3 × y = 210 (Use the horizontal format)

Notice that unless we have different colors or write very neatly, we can no longer identify what the original problem was!

Next solve the same problem with the vertical format.

Example 16

Solve 3 × y = 210 (Use the vertical format)

This takes a little more space, but we can identify what the original problem was. This helps us when we finish a test and go back to check our work. Later, in future math classes, this format also helps avoid careless errors in more complicated problems.

Here is another vertical format sample:

Example 17

Solve 9 × y = 189 (Use the vertical format)

I used both black and blue writing. In Math 20, write a lot, like I did. But in Math 60 and 65 one goal will be to eventually wean yourself from always using the steps I did in blue. The instructor will write fewer of these steps.

There are two other reasons to use the vertical format.

First, it promotes doing homework in two columns per page. This often saves paper. Homework problems, like math lecture notes, by their nature are seldom as wide as a page.

Second, putting work in that shape makes it easier to do scratch work off on the side. Watch how that helps me stay organized when solving for y when fraction arithmetic happens.

Example 18

Solve 8 × y = 23 (Use the vertical format)

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Chapter 1 Test, Problem 28: Solve: 28 + x = 74

Chapter 1 Test, Problem 29: Solve: 169 ÷ 13 = n

Chapter 1 Test, Problem 30: Solve: 38 × y = 532

Chapter 1 Test, Problem 31: Solve: 381 = 0 + a

Chapter 2 Test, Problem 34: Solve: 78 × x = 56

Chapter 2 Test, Problem 35: Solve: t × 25 = 710

Chapter 3 Test, Problem 9: Solve: 14 + y = 4

Chapter 3 Test, Problem 10: Solve: x + 23 = 1112

Chapter 4 Test, Problem 32: Solve: 4.8 × y = 404.448

Chapter 4 Test, Problem 33: Solve: x + 0.018 = 9

Problem Solving Grammar

Consider again the problem 3 × y = 210. Notice that there are many possible ways to write the step of dividing both sides by 3. The clearest is to use the vertical format and write either ÷ 3 or /3 on its own line. A few possibiltiies are considered incorrect because of bad math grammar.

Example 19

What are some incorrect ways to write the step of dividing both sides by 3 when solving 3 × y = 210?

(a) Do not use brackets to incorrectly mean "do this to the entire equation", as in ÷ 3 (3 × y = 210).

(b) Do not use brackets on each side of the equation improperly, as in ÷ 3 (3 × y) = (210) ÷ 3. Notice the right hand side is legitimate. But the left hand side begins confusingly with the ÷ symbol.

In a future math class studying algebra you will encounter other incorrect ways. For example, when solving 3 × y + 3 = 210 it would be smart to divide both sides by 3 but it would violate the distributive property to incorrectly write 3 × y + 3 ÷ 3 = 210 ÷ 3

Shoes and Socks link to here link to index

When you get dressed, you put your socks on before your shoes. To undress you must remove your shoes first, because they were put on most recently.

The same reversal happens when isolating a variable.

The Shoes and Socks Theorem

When solving for a variable, the number attached last to the variable needs to be removed first.

This sounds funny, but makes sense when you see it happen.

Example 20

Solve 30 = 5 × y − 10

Consider the order of operations.

Multiplication has priority over subtraction. So the 5 × has first priority and the − 10 has last priority.

First remove the − 10.

30 = 5 × y − 10

+10 +10

40 = 5 × y

Then remove the 5 ×.

40 = 5 × y

÷5 ÷5

8 = y

The Shoes and Socks Theorem can help deal with negatives. Remember that a number is made negative by × (−1).

Example 21

Solve 25 − y = 20

Consider the order of operations.

Multiplication has priority over subtraction. So the × (−1) has first priority and the positive 25 has last priority.

First remove the 25. It is a positive number, so remove it using subtraction.

25 − y = 20

−25 −25

y = − 5

Then remove the × (− 1). Because "a negative of a negative is a positive" we can do that be repeating it.

y = − 5

×(−1) ×(−1)

y = 5

Often it helps to think about terms. For example, with the equation 90 = 7 × y − 11 × 2. we know the first step is to simplify the 11 × 2 term so it will be easier to remove from the right hand side of the equation. Once we have 90 = 7 × y − 22 the equation is as simple as Example 7 above.