PRIME AND COMPOSITE NUMBERS

FACTORS AND MULTIPLES

 

Everywhere we turn we see numbers...on the telephone, houses, clocks, sizes , pages , money, etc. There's no escaping them! We use numbers for just about everything in our lives. Of course, one of the first things we learn is to count...one, two, buckle my shoe; three, four, shut the door, and so on. Our knowledge of numbers grows to adding money ...ummm, three dimes or 6 nickels can buy me a bag of M&M's!

Communicating information using numbers is a must in this world; at work, at everyday living, as well as at play... What's the measurement of the new cabinet you just built?...How many cups of milk do I need to make 2 cakes?...How many yards do we have to run to make that touchdown?

Thinking about other things in our lives in terms of numbers just seems to happen...Why does New Years Day fall on a different day of the week each year...Why 30 days in some months, 31 days in others, and let's not forget February and leap year!

Using numbers to help us solve problems is the best part! This task is sometimes easier said than done. The more we learn about the properties of numbers, the better we become at using them. The more we learn, the more questions we should have and the more answers we receive, the better our understanding becomes.

So, don't be afraid to ask! For instance....

Why can some numbers be divided by lots of numbers and others only by a few?

 

 


EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT WHOLE NUMBERS BUT WERE AFRAID TO ASK!

LET THE GAMES BEGIN...

10,9,8,7,6,5,4,3,2,1,0

 

PRIME NUMBER OR COMPOSITE NUMBER ?

THAT IS THE QUESTION!

 

PICK A NUMBER...ANY NUMBER FROM 2 TO 10

ME? ...........I PICK "3"

What do I know about this number?

a. Adding 2 & 1 = 3; or, 1 & 2 = 3; or 3 & 0 = 3; or, 1 & 1 & 1 = 3

b. Subtracting 3 - 3 = 0; 4 - 3 = 1; 5 - 3 = 2; 10 - 7 = 3; and so on...

c. Multiplying 1 x 3 = 3; or 3 x 1 = 3...any more?

d. Dividing 3 by 1 = an even 3; or, 3 by 3 = an even 1...that's it?!

e. I know that in baseball...3 strikes and yurrrrr out.

f. 3 is an odd number.

3 IS CALLED A PRIME NUMBER BECAUSE IT CAN BE DIVIDED EVENLY BY ONLY 1 AND ITSELF.

ANOTHER WAY TO RECOGNIZE IF A NUMBER IS PRIME IS IF IT HAS ONLY TWO FACTORS, 1 AND ITSELF. WELL...WHAT'S A FACTOR?!?

A FACTOR IS ONE OF TWO OR MORE NUMBERS THAT ARE MULTIPLIED TO GET A PRODUCT.

1 x 3 = 3

1 is a factor and 3 is a factor of the product 3

 

WHAT ARE THE PRIME NUMBERS BETWEEN 0 AND 12?

0,1,2,3,4,5,6,7,8,9,10,11,12

 

 

 

 

WHAT IS THE ONLY EVEN PRIME NUMBER?

0,1,2,3,4,5,6,7,8,9,10,11,12

 

 

 

Don't be fooled though...all other even numbers are composite but all odd numbers are not prime!

Can you pick out any odd numbers that are composite?

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25

What numbers from the above list are prime?

GREAT JOB...

YOU'RE BECOMING AN EXPERT!

FYI

0 and 1 are not considered prime or composite.

No prime number greater than five ends in a five.

A number can not be both prime and composite.

 

 

 

NOW PICK ANOTHER NUMBER...please

YOU PICK "6"

What do you know about this number?

a. Adding 2 & 4 = 6; what others add up to 6?

b. Subtracting 6 - 6 = 0; 6 - 5 = 1; 6 - 4 = 2; 13 - 5 = 6; what else?

c. Multiplying 6 x 1 = 6; 2 x 3 = 6;

WOW IT HAS 4 FACTORS! 1, 6, 2 AND 3

d. Dividing 6 by 1 = 6; 6 by 3 = 2;

6 by 2 = 3; 6 by 6 = 1

e. 6 points in basketball is like getting 3 baskets at 2 points each...SWISSSHHH!

f. 6 is an even number.

How many regular baskets did the team get if the score was 12 at the end of the first quarter?

 

OK, THE $64,000 QUESTION IS...WHAT KIND OF NUMBER IS 6...?

IT CANNOT BE PRIME BECAUSE IT HAS MORE THAN TWO FACTORS

THE FACTORS OF 6 ARE 1,2,3,AND 6

The FACTOR PAIRS are:

1 X 6 and 6 X 1

2 X 3 and 3 X 2

 

THE NUMBER 6 IS CALLED A COMPOSITE NUMBER BECAUSE IT HAS MORE FACTORS THAN 1 AND ITSELF; OR, IT CAN BE DIVIDED EVENLY BY MORE NUMBERS THAN ONE AND ITSELF.

My favorite definition of a COMPOSITE NUMBER is any number, greater than 1, that is not prime.

 

(Remember, neither 0 nor 1 is prime or composite.)

0,1,2,3,4,5,6,7,8,9,10,11,12

 

What are the COMPOSITE NUMBERS between 0 and 12?

What is the first COMPOSITE NUMBER we come to counting from 2?

 

? X 5 = 15

What factor is paired with 5 to give 15?

What are all the factor pairs of 16?

HINT- Start with the number 2 and ask yourself,

"Does 2 go into 16 evenly?"

Of course the answer is YES because every even number is divisible by 2.

Then ask about 3, then 4, then 5 and so on.

 

LET'S PLAY "I'M THINKING OF A NUMBER"

Clue 1.....The digits of my number total 1.

Clue2.....My number has two digits, neither is prime nor composite.

Clue 3.....My number has 4 factors.

Clue 4.....The difference between the two digits of my number is 1.

 

ARE YOU SEEING ANY CONNECTION BETWEEN FACTORS &

PRODUCTS AND DIVISORS, QUOTIENTS & DIVIDENDS IN

DIVISION?

SEE IF YOU NOTICE ANYTHING HERE....

1 x 10 = 10

10 DIVIDED BY 1 = 10

2 X 5 = 10

10 DIVIDED BY 5 = 2

10 DIVIDED BY 2 = 5

*factor X factor = product.

*The product divided by one of the factors of a factor pair will always equal the other factor as a quotient.

 

You try the number 18.

What are all the connections you can make with the factors and products...divisors and quotients?

 

CHALLENGE

The formula for the area of a rectangle...A = Length x Width

 

What are all the possible dimensions, (measurements of the sides), of a rectangle with an area equaling 36 square feet?

HINT- The area is like the product and the dimensions are like the factor pairs...good luck!

 

 

GREAT JOB!

 


 

DIVISIBILITY RULES

 

Believe it or not....

There is a quick way to determine what some of the factors of a number are without actually doing the division.

Boy is that good news....we can use all the help we can get.

Try to remember some of these facts.....

 

A NUMBER IS DIVISIBLE BY

 

2 if...it ends in a 0,2,4,6 or 8.

 

3 if...the sum of its digits is divisible by 3. For example, the number 135. Add its digits, 1+3+5=9. Yes, 135 is divisible by 3 because the sum of its digits, 9, is divisible by 3.

 

4 if...the last two digits are divisible by 4 or it ends in 00. 632...what do you think? YES!

 

5 if...the last digit is a 5 or 0. 25,30,35,40,45,and so on and so on...

 

6 if...the number is divisible by 2 and 3. For example...12,18,24 and many more! Can you think of any?

 

8 if... the number formed by the last two digits is divisible by 8.

 

9 if...the sum of the digits is divisible by 9.

 

10 if...the number ends in a 0. How easy is that!!!!!

 

 


 

Now you are ready for MULTIPLES!!!

A multiple is the product of a whole number and any other whole number.

For example, 6 is a multiple of 3,

because 2 x 3 = 6.

 

To figure out the next multiple of 3 use the next number...how easy....

3 x 3 = 9 and 4 x 3 = 12 and 5 x 3 = 15

So, let's check this out...9, 12 and 15 are all multiples of 3.

 

OK

 

Are you ready for this?

 

What are the first four multiples of the number 4?

Can you find any multiples that the numbers 3 and 4 share?

WHEN NUMBERS SHARE A MULTIPLE IT IS CALLED A COMMON MULTIPLE.

Hint...The multiples of 3 are, 3,6,9,12,15,18,21,24 and so on....

and

The multiples of 4 are 4,8,12,16,20,24,28 and so on...

 

Yes, the numbers, called common multiples, are 12 and 24. What do you predict the next common multiple will be for 3 and 4?

How did you know? What have you discovered?

A quick review.

The factors of 12 are 1,2,3,4,6, and 12

the factors of 28 are 1,2,4,7, and 14, and 28

 

So, the common factors of these two numbers are 1,2,and 4.

A COMMON FACTOR IS A FACTOR THAT TWO OR MORE NUMBERS SHARE.

 

 


PRIME FACTORIZATION

 

Prime Factorization is a way to express a number. Any number is the product of prime numbers that have been multiplied.

WHAT AM I TALKING ABOUT????

 

Well, this is definately the kind of math fact that you want to see an example

of ...and fast!

 

Give me a number between 2 and 50

 

"48" Good Choice

What are the factors of 48?

(quick review)

The factor pairs are...2x24,3x16,4x12,6x8

So, the factors are 2,3,4,6,8,12,16,24

 

Now none of these factor pairs are the Prime Factorization because each pair includes one or more composite number as a factor.

BUT

If I keep dividing up the number 48 using prime numbers,

one of the outcomes would look like this...

48 divided by 2 = 24

24 divided by 2 = 12

12 divided by 3 = 4

4 divided by 2 = 2

If I take all the prime numbers that I used for dividing up 48, ( they are highlighted in red ), and I multiply them, the product will be 48!!!

2 x 2 x 3 x 2 = 48

or

2 x 2 x 2 x 3 = 48

or

3 x 2 x 2 x 2 x = 48

It does not matter what order you multiply these prime numbers...you will always get 48.

 

Most Importantly!!!!!

No matter how you choose to Prime Factor the number 48, you will ALWAYS GET THE SAME FACTORIZATION AS AN ANSWER!

 

Challenge

 

Choose a number with at least four factor pairs and find four different ways to get the prime factorization. Do you end up with the same prime factorization each time?

If you do....YOU ARE NOW A MASTER FACTOR WIZARD!

!!!CONGRATULATIONS!!!