Download 5.1 Divisibility and Primes INSTRUCTOR NOTES

5.1 Divisibility and Primes
INSTRUCTOR NOTES
1 HOUR 15 MIN.
Ask students their definition of a “prime number”.
p. 298 A PRIME NUMBER is an integer greater than 1 whose only
positive factors are 1 and itself.
An integer greater than 1 that is not prime is a COMPOSITE NUMBER.
The numbers 2 and 11 are prime because the only way you can factor them is
2 = 1*2
11= 1*11
The numbers 6 and 175 are composite numbers because
6 = 2*3
175 = 5*35
Write on Board: You and the students together derive all of the prime
numbers up to 50 (write them on the side board as you do this).
The primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
p. 299
Reference The Fundamental Theorem of Arithmetic
Note that 175 can be factored further into only prime factors as such
175 = 5*5*7 = 5 *7
This is the prime factorization of 175.
The prime factorization of a prime number is just the number itself. For
example, the prime factorization of 7 is just 7.
1
You do
Find the prime factorization of 3150.
ANSWER: 3150
2*1575
2*5*315
2*5*5*63
2*5*5*7*9
2*5*5*7*3*3
So
3150 = 2 ∙ 3 ∙ 5 ∙ 7
Work can also be represented as follows
3150
/ \
2 1575
/ \
5 315
/ \
5 63
/ \
7 9
/ \
3 3
So 3150 = 2 ∙ 3 ∙ 5 ∙ 7
CLASS DO: Find the prime factorization of 18.
Answer: 18 = 2 ∙ 3
CLASS DO: Find the prime factorization of 408.
Answer: 408 = 2 ∙ 3 ∙ 17
CLASS DO: Find the prime factorization of 2002.
Answer: 2002 = 2 ∙ 7 ∙ 11 ∙ 13
2
CLASS DO: Find the prime factorization of 2011.
Answer: 2011
Note that 2011 is prime.
Write on Board:
How far should you go when testing to see if a number is prime? Only up to √
where is the number to be tested. Reference p.301 Test for Primality.
Therefore, when determining if 2011 is prime you only have to divide it by primes
up to √2011 ≈ 44.8. So you don’t need to check past 43.
CLASS DO: Check to see if 3001 is prime.
Answer: It is prime. Partial work: √3001 ≈ 54.8
CLASS DO: Find the prime factorization of 1970.
Answer: 1970 = 2 ∙ 5 ∙ 197
p. 305
Reference The Division Algorithm
= ∙ +
with 0 ≤ <
is the number you are dividing by
is called the “quotient”
is called the “remainder”
Complete the following table on the board showing steps done on the calculator.
Division
Quotient
Remainder
= ∙ +
÷
q
r
17÷3
5.667
5
17-5*3=
2
17=5*3+2
28÷11
2.545
2
28-2*11=
6
28=2*11+6
3055÷123 24.837
24
3055-24*123=
103
3055=24*123+103
21÷3
7
7
21-7*3=
0
21=7*3+0
2÷8
.25
0
2-0*8=
2
2=0*8+2
-17÷3
-5.667
-6
-17-(-6*3)=
1
-17=-6*3+1
-55÷12 -4.583
-5
-55-(-5*12)=
5
-55=-5*12+5
-247÷19
-13
-13
-247-(-13*19)=
0
-247=-13*19+0
3