Math 6A (Periods 2 & 4)

Tests for Divisibility 5-2

It is important to learn the following divisibility rules:
A number is divisibility by:

2 ... if the ones digit of the number is even
3 ... if the sum of the digits is divisible by three ( add the digits together)
4 ... if the number formed by the last two digits is divisible by by four ( Just LOOK at the last two numbers-- DON"T ADD them!!)
5 ... if the ones digits of the number is a 5 or a 0
6 ... if the number is divisible by both 2 and 3... (or if it is even and divisible by 3)
8 ... if the number formed by the last three digits is divisible by 8. (Like FOUR, just look at the last three digits-- divide them by 8)
9 ... if the sum of the digits is divisible by 9
10 ... if the ones digits of the number is a 0.

You will not need to know the divisibility rules for 7 or 11 but they are interesting...

You can test for divisibility by 7
Let's start with a number 959
Step 1: drop the one's digit so we have 95
Step 2: Subtract twice the ones' digit ( that you dropped) in this case we dropped a 9
so we double that and subtract 18 from 95
or 95-18 = 77. If the results, in the case, 77, is divisible by 7 --- so is the original number 959.
Step 3: If the number you get is still to big.. continue the process until you can determine if your number is divisible by 7.


To test for divisibility by 11
add the alternative digits beginning with the first
so let's try the following
4,378,396
Step 1: Add the alternate digits beginning with the 1st 4 + 7+ 3 + 6 = 20
Step 2: Add alternate digits beginning with the 2nd 3 + 8 + 9 = 20

Step 3: If the difference of the sums is divisible by 11 so is the original number.
In this case, 20-20 = 0 and 0/11= 0 so
4,378,396 is divisible by 11.


A good test for divisibility by 25 would be if the last two digits represent a multiple of 25.

A perfect number is one that is the SUM of all its factors except itself. The smallest perfect number is 6, since 6 = 1 + 2+ 3
The next perfect number is 28 since
28 = 1 + 2 + 4 + 7 + 14
What is the next perfect number?

Algebra Honors ( Period 5 & 6)

Using Factoring to Solve Problems  5-13

Example 1

A decorator plans to place a rug in a 8 m by 12 m room so that a uniform strip of wood flooring around the rug will remain uncovered. How wide will this strip be if the area of the rug is to be half the area of the room?

Let x = the width of the strip

Then 12-2x is the length of the rug and 9 – 2x is the width of the rug

(12-2x)(9-2x) = the area of the rug.

Area of the rug = ½ ( area of the room)

(12-2x)(9-2x) = ½(9∙12)

108 – 42x + 4x² = 54

4x²- 42 + 108 = 54

4x²- 42 -54 = 0

2(2x²- 21 + 27) =0

2(2x -3)(x -9) = 0

using the ZERO Products Property

2x -3 = 0 or x= 3/2

x -9 = 0  x = 9

CHECK: x = 1.5

works but when x = 9

the length 12 – 2x and the width 9-2x are negative! Since a negative length or width is meaningless reject x = 9 as an answer

This show that also the equation has a root that does not check because this equation does not  meet the hidden requirements that the rug have a positive length.  

Example 2
The FORMULA h = rt – 4.9t² is a good approximation of the height (h) in meters of an object t seconds after it is projected upward with an initial speed of r meters per second.

An arrow is shot upward with an initial speed of 34.3 m/s. When will it be at a height of 49 m?

let t = the  number of seconds  after being shot that the arrow is 49 m high.

Let h = the height of the arrow= 49 m

Let r = the initial speed = 34.3 m/s

Substitute in the formula

h = rt – 4.9t²

49 = 34.3t – 4.9t²

4.9t² – 34.3t + 49 = 0  THINK—GCF???

4.9(t² – 7t + 10 ) = 0

4.9(t -2)(t -5) = 0

Using the ZERO PRODUCTS PROPERTY

t = 2 and t = 5

Therefore the arrow is 49 m high both 2 seconds and 5 seconds after being shot… on its way up and on its way down!

Example 3

If a number is added to its square, the results is 56. Find the number

 Let x = the number

x² + x = 56

(x +8)(x - 7) = 0

 x = -8 and x = 7

Find two consecutive positive odd integers whose product is 143

Let x = the first positive odd integer

Let x + 2 = the 2^nd positive odd integer

x(x + 2) = 143

x² + 2x = 143

x² + 2x – 143 = 0

(x + 13)(x – 11) = 0

x = -13 and x = 11

But ask only for the positive integers so reject -13

The two integers are 11 and  13

Example 4

The sum of the squares of two consecutive negative odd integers is 290. Find the integers

let x = the 1^st negative odd integer

let x + 2 = the 2^nd negative odd integer

x² + (x + 2)² = 290

x² + x² + 4x + 4 = 290

2x² + 4x – 286 = 0  THINK  GCF!!!

2(x²+ 2x -143) = 0

2(x + 13)(x -11) = 0

x = -13 and x = 11

But it ask for the negative integers  so reject x = 11

The two negative integers are -13 and -11

Math 6A (Periods 2 & 4)

Finding Factors and Multiples 5-1


You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 60. A number is said to be divisible by its whole numbered factors.

To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.

We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6

When you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!

A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first five multiples of 7 are
0, 7, 14, 21, 28
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21; 4(7) = 28
... and put in set notation it would be
{0, 7, 14 ,21, 28}



If you were to ask for the first four NON-ZERO Multiples of 6
the answer would be 6, 12, 18, 24.. and in set notation
{ 6, 12, 18, 24}
Whereas the first four multiples of 6 ( you would need to include 0)
{0, 6, 12, 18}



Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!

Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number

What number is a factor of every number? ONE
Is every number a factor of itself? YES
What is ( are) the only multilpe(s) of 0? 0
How many numbers have 0 as a factor? only one number What number(s)? ZERO



The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.