Download Prime Numbers and Prime Factorization

13
Tallahassee Community College
PRIME NUMBERS AND FACTORING
(Use your math book with this lab)
I.
Divisors and Factors of a Number
Previously, you learned the names of the parts of a multiplication
problem.
1.
a.
6 × 2 = 12
b.
6 and 2 are the___________________
12 is the_________________________
You learned the names of the parts of a division problem from your
textbook.
2.
a.
b.
6
2)12
12
1
8)12
-8
4
2 is the_________________________
12 is the_________________________
6 is the_________________________
the remainder is__________________
8
12
1
4
is
is
is
is
the_________________________
the_________________________
the_________________________
the_________________________
The definitions of a factor of a certain whole number or the divisor of
the whole number are in your textbook. The factors or divisors of a number
will be natural numbers (1, 2, 3, 4, ...). REMEMBER we cannot divide by
zero.
3.
a.
b.
c.
d.
e.
What is the smallest factor of 6?__________________
What is the largest factor of 6? __________________
What is the smallest factor of 27?_________________
What is the largest factor of 27? _________________
Generally speaking, the smallest factor of any number
is______________________; the largest factor is
________________________.
II.
Finding All Factors of a Number (Study your textbook.)
Now study these ways of writing a number as a product of two natural
numbers:
│
│
1 × 10 = 10
│
2 × 5 = 10
│
3 ×
│
4 ×
│
5 × 2 = 10
│
6 ×
│
7 ×
│
8 ×
│
9 ×
│
10 × 1 = 10
│
│
The factors
│
of 10 are 1, 2, │
5, 10
│
│
│
10
II.
1.
b.
c.
The factors of 16 are 1, 2, 4, 8
and 16.
List all________________numbers, stopping with the first
number whose square is ____________ to or_______________
than 20.
__________ by each of these numbers.
The factors of 20 are both the divisors and quotients of
the divisions which have reminders of__________________.
36
b.
42
c.
37
d.
20
A number is divisible by 2 if__________________________
A number is divisible by 3 if__________________________
A number is divisible by 5 if__________________________
Look at the factors of 37.
a.
5.
You can see that it is not
necessary to try all natural
numbers. You can stop after
you try the first factor
whose square is equal to or
greater than the original
number.
Your work will be easier if you learn the divisibility rules
in your textbook.
a.
b.
c.
4.
× 16 = 16
× 8 = 16
×
× 4 = 16
×
×
×
× 2 = 16
×
×
×
×
×
×
×
× 1 = 16
Find all divisors of each number.
a.
3.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Tell how you would find all the factors of 20.
a.
2.
16
There are exactly 2 different factors of 37; what are
they?_________, __________.
b. 37 is a ______________number because its only factors
are 1 and the number itself.
Look at the factors of 36.
a.
b.
6.
Are 1 and 36 its only factors?_______________________
What kind of number is 36?___________________________
Find all prime numbers less than 20._____________________
III. The Prime Factorization of a Number.
It is very important that you know what a prime factorization of a
number is. A factorization of a number is a multiplication problem whose
product is the number.
a.
b.
c.
1 × 8
2 × 4
2 × 2 × 2
All of these are factorizations of 8 because
the product is 8 in each case.
NOTICE all of the factors in "c" are prime numbers.
the prime factorization of 8.
We say 2 x 2 x 2 is
To find the prime factorization of a number, divide by prime numbers
only. (REMAINDERS MUST BE ZERO)! Continue until the quotient is a prime
number.
The prime factorization is the expression which shows all of the prime
divisors and the final prime quotient as factors.
Study the prime factorization of 36.
(Compare this with 2a in Part II above.)
3
3) 9
2)18
2)36
2 × 2 × 3 × 3 is the prime factorization of 36.
Students sometimes miss test questions because they aren't sure what is
Earlier, you learned to find all
meant.
Don't let this happen to you.
factors of a number and you learned to find the prime factorization of a
number.
In 1 and 2 below, you answer the question.
In 3 and 4, you write the question so that the answer is correct.
1.
Find the prime factorization of 70.
(Your answer must be a multiplication problem in which all
factors are primes.)
2.
Find all the factors of 70.
(This answer is a list of all the numbers that divide evenly
into 70. The numbers are separated with commas.)
3 and 4. Study.
3.
Find_______________________________ of 45.
WORK 1 × 45
2
3 × 15
4
5 × 9
6
7
4.
WORK
5.
Then tell what was found!
ANSWER: 1, 3, 5, 9, 15, 45
Find_______________________________ of 45.
5
3)15
3)45
ANSWER:
3 × 3 × 5
Find the prime factorization of each composite number.
If the number is prime, write "PRIME".
a. 50
b. 48
c. 27
d. 29
ANSWERS:
I.
1.
2.
3.
II.
1.
2.
3.
4.
a. 6, 2 are factors
b. 12 is the product
a. 2 is divisor
12 is dividend
6 is quotient
0 is remainder
b. 8 is divisor
12 is dividend
1 is quotient
4 is remainder
a. 1
b. 6
c. 1
d. 27
e. smallest factor is 1; largest is the number itself.
a.
b.
c.
a.
b.
c.
d.
a.
b.
c.
a.
b.
natural, equal, greater
divide
zero
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
42: 1, 2, 3, 6, 7, 14, 21, 42
37: 1, 37
20: 1, 2, 4, 5, 10, 20
last digit is 0, 2, 4, 6, or 8
sum of digits is divisible by 3
last digit is 0 or 5
1, 37
prime
Answers (continued)
II. (continued)
5.
a. No
b. composite
6.
III. 1.
2.
2, 3, 5, 7, 11, 13, 17, 19 (One is not prime. It has
only one factor.)
2 × 5 × 7
1, 2, 5, 7, 10, 14, 35, 70
3.
All the factors
4.
Prime factorization
5.
a.
2 × 5 × 5
b.
2 × 2 × 2 × 2 × 3
c.
3 × 3 × 3
d.
prime