section 2.1 – factors and multiples · only whole number that is neither prime nor composite....
TRANSCRIPT
Copyright © 2014 – Luis Soto-Ortiz 104
Section 2.1 – Factors and Multiples
When you want to prepare a salad, you select certain ingredients (lettuce,
tomatoes, broccoli, celery, olives, etc.) to give the salad a specific taste. You can
think of the salad as the final result, or product, of combining the ingredients. In
general, the order in which you combine these ingredients does not affect the
taste and nutritional value of the salad.
We saw in the previous chapter that whenever we multiply two or more whole
numbers, we get another whole number as the final result. For example,
× × =
We can think of the numbers 17, 3 and 11 as the “ingredients” that multiplied
together make up the “salad,” or final result of 561. In mathematics, the
ingredients are called factors and we call the salad a product or multiple. Thus,
we say that the whole numbers 17, 3 and 11 are the factors that multiplied
together give the product 561. The number 561 is a multiple of 17, 3, and 11,
which means that 561 is divisible by 17, 3, and 11 because $561 can be evenly
divided among 17 people, among 3 people or among 11 people without resorting
to cents.
In the following equation, name the factors and the multiple.
× × × = ,
Example 2.1.1
Answer: The numbers 13, 47, 5 and 241 are factors of the product 736,255,
whereas the product 736,255 is a multiple of 13, 47, 5 and 241. This means that
736,255 is divisible by 13, 47, 5 and 241. You can think of 736,255 as being the
“salad” and the numbers 13, 47, 5 and 241 the “ingredients” that multiplied
together give 736,255.
Copyright © 2014 – Luis Soto-Ortiz 105
In the following equation, name the factors and the product.
× × × = ,
Note: The order in which you multiply the factors doesn’t affect the final product
because multiplication is commutative.
For example: 8 × 5 × 2 = 5 × 8 × 2 = 2 × 8 × 5 = 80
Whenever we can write a whole number as the product of a set of whole
numbers (its factors), we say that the product is divisible by those numbers. That
is, any whole number is divisible by its factors.
Since 7 × 8 = 56, we know that 56 is divisible by 7 and that 56 is divisible by 8.
Therefore, 56 is a multiple of 7 and 8, while 7 and 8 are factors of 56. The
complete list of factors of 56 is 1, 2, 4, 14, 28 and 56.
8
7 5 6
- 5 6
0
7
8 5 6
- 5 6
0
Example 2.1.2
Answer: The numbers 6, 4, 23 and 11 are factors of 6,072, whereas the number
6,072 is a multiple of 6, 4, 23 and 11.
Example 2.1.3
Since 7x8=56, then 56 is the product and 7
and 8 are factors of 56. Notice that
whenever we divide a product by one of its
factors, the quotient is also a factor and the
remainder is always zero.
factors multiple
factors multiple
Copyright © 2014 – Luis Soto-Ortiz 106
Since 2 × 28 = 56, we know that 56 is divisible by 2 and that 56 is divisible by 28.
Therefore, 56 is a multiple of 2 and 28, while 2 and 28 are factors of 56.
Note: 1 is a factor of any whole number because 1 × = . This also means
that any whole number is divisible by 1.
For example,
1 is a factor of 17 since 1 × 17 = 17. Therefore, 17 is divisible by 1.
1 is a factor of 5,788 since 1 × 5,788 = 5,788. Therefore, 5,788 is divisible by 1.
Let’s check that the remainder is zero:
2 8
2 5 6
- 5 6
0
2
2 8 5 6
- 5 6
0
1 7
1 1 7
- 1
0 7
- 7
0
5 7 8 8
1 5 7 8 8
- 5
0 7
- 7
0 8
- 8
0 8
- 8
0
Factor
Factor Multiple
- :
0
Example 2.1.4
Since 2x28=56, then 56 is the product and
2 and 28 its factors. Notice that whenever
we divide a product by one of its factors,
the quotient is also a factor and the
remainder is always zero.
remainder
factors
multiple factors
multiple
Copyright © 2014 – Luis Soto-Ortiz 107
The following are special whole numbers that you should become familiar with:
Number Definition Examples
Even Any number that is divisible by 2. It has 2
as a factor.
0,2,4,6,8,10,12,14,16, …
Odd Any number that is not divisible by 2. It
does not have 2 as a factor.
1,3,5,7,9,11,13,15,17,…
Prime Any whole number that has exactly 2
different factors: 1 and the number itself.
2,3,5,7,11,13,17,19,23,...
Composite Any whole number greater than 1 that is
not a prime number.
4,6,8,9,10,12,14,15,16,…
Note: The number 1 is not a prime number because it has only one factor (itself),
since 1 × 1 = 1. Moreover, 1 is not a composite number either. In fact, 1 is the
only whole number that is neither prime nor composite.
Table of Prime Numbers Less Than 1,000
2 3 5 7 11 13 17 19 23
29 31 37 41 43 47 53 59 61 67
71 73 79 83 89 97 101 103 107 109
113 127 131 137 139 149 151 157 163 167
173 179 181 191 193 197 199 211 223 227
229 233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337 347
349 353 359 367 373 379 383 389 397 401
409 419 421 431 433 439 443 449 457 461
463 467 479 487 491 499 503 509 521 523
541 547 557 563 569 571 577 587 593 599
601 607 613 617 619 631 641 643 647 653
659 661 673 677 683 691 701 709 719 727
733 739 743 751 757 761 769 773 787 797
809 811 821 823 827 829 839 853 857 859
863 877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997
Copyright © 2014 – Luis Soto-Ortiz 108
Make a table of all the factors of 24.
The number at the top of the table shown in blue (24) is a multiple of all the
numbers in green. Conversely, the numbers in green are factors of 24.
In the table above, we see that 24 is a factor and multiple of itself. In fact, any
whole number is a factor and multiple of itself because we can always write
1 × = .
Make a table of all the factors of 120.
Example 2.1.5
Answer:
24
1 24
2 12
3 8
4 6
1, 2, 3, 4, 6, 8, 12 and 24
are factors of 24 because
1x24=24
2x12=24
3x8=24
4x6=24
Example 2.1.6
Answer:
120
1 120
2 60
3 40
4 30
5 24
6 20
8 15
10 12
1, 2, 3, 4, 5, 6, 8, 10, 12,
15, 20, 24, 30, 40, 60 and
120 are factors of 120
because
1x120=120
2x60=120
3x40=120
4x30=120
5x24=120
6x20=120
8x15=120
10x12=120
Copyright © 2014 – Luis Soto-Ortiz 109
Make a table of all the factors of 700.
The number 700 at the top of the table (shown in blue) is a multiple of all the
numbers in green. Similarly, the numbers in green are factors of 700.
Please seek help from your instructor if you have difficulty understanding the
difference between a multiple and a factor.
Here is an instructional video on what a multiple of a number is:
http://www.youtube.com/watch?v=vbNeXLvqM90
Example 2.1.7
Answer:
700
1 700
2 350
4 175
5 140
7 100
10 70
14 50
20 35
25 28
1, 2, 4, 5, 7, 10, 14, 20,
25, 28, 35, 50, 70, 100,
140, 175, 350 and 700 are
factors of 700 because
1x700=700
2x350=700
4x175=700
5x140=700
7x100=700
10x70=700
14x50=700
20x35=700
25x28=700
Copyright © 2014 – Luis Soto-Ortiz 110
1. Is 4 a factor of 216? Explain why or why not.
2. Is 17 a factor of 91? Explain why or why not.
3. Given that 5 × 13 × 17 = 1105 state whether the following statements are
true or false:
A. 5 is a factor of 1105.
B. 17 is a factor of 1105.
C. 1105 is a factor of 13.
D. 1105 is a multiple of 17.
E. 1105 is a multiple of 5.
F. 1105 is divisible by 13.
G. 5 is a factor of 13.
4. Is 200 a multiple of 10?
5. Is 512 a multiple of 16?
6. Is 720 a multiple of 11?
7. Is 77 a multiple of 11?
8. Is 11 a multiple of 77?
9. Write a table of all the factors of 90.
10. Write a table of all the factors of 64.
11. True or false:
a. 7 is a prime number.
b. 90 is a prime number.
Classwork 2.1
Copyright © 2014 – Luis Soto-Ortiz 111
c. 63 is a composite number.
d. 15 is a prime number.
e. 41 is a composite number.
f. 28 is a composite number.
g. 9 is a prime number.
h. 1 is a prime number.
i. 1 is a composite number.
j. 945 is a composite number.
k. 121 is a prime number.
l. 200 is a composite number.
12. Write a table of all the factors of 30.
13. True or false:
a. 31 is a prime number.
b. 70 is a composite number.
c. 81 is a composite number.
d. 9 is a prime number.
e. 596 is a composite number.
f. 45 is a prime number.
g. 111 is a prime number.
h. 441 is a prime number.
i. 3,250 is a composite number.
j. 777 is a composite number.
Copyright © 2014 – Luis Soto-Ortiz 112
k. 169 is a prime number.
l. 1,011 is a prime number.
14. T/F _____ 90 is divisible by 10.
15. T/F _____ 648 is divisible by 5.
16. T/F _____ 7,326 is divisible by 3.
17. T/F _____ 916 is divisible by 4.
18. T/F _____ 5,783 is divisible by 2.
19. T/F _____ 3,111 is divisible by 3.
20. T/F _____ 12 is divisible by 8.
Copyright © 2014 – Luis Soto-Ortiz 113
4 216 4 × 54 = 216 ( 216 4 0 ).
17 91 91 17 .
CW 2.1 Solutions:
1.
2.
3. A. True B. True C. False D. True E. True F. True G. False
4. Yes 5. Yes 6. No 7. Yes 8. No
9. 10.
11. a. True b. False c. True d. False e. False f. True g. False h. False i. False
j. True k. False l. True
12.
13. a. True b. True c. True d. False e. True f. False g. False h. False i. True j. True
k. False l. False
14. T 15. F 16. T 17. T 18. F 19. T 20. F
90
1 90
2 45
3 30
5 18
6 15
9 10
64
1 64
2 32
4 16
8 8
30
1 30
2 15
3 10
5 6
Copyright © 2014 – Luis Soto-Ortiz 114
1. Is 9 a factor of 325? Explain why or why not.
2. Is 12 a factor of 240? Explain why or why not.
3. Given that 8 × 19 = 152 state whether the following statements are true or
false:
A. 8 is a factor of 152.
B. 19 is a factor of 152.
C. 152 is a multiple of 8.
D. 152 is divisible by 19.
E. 152 is a multiple of 19.
F. 152 is a factor of 8.
G. 8 is a factor of 19.
4. Is 34,590 a multiple of 5?
5. Is 400 a multiple of 1,200?
6. Is 6 a factor of 30?
7. Is 144 divisible by 8?
8. Is 341 divisible by 3?
9. Write a table of all the factors of 44.
10. Write a table of all the factors of 150.
11. True or false:
a. 33 is a prime number.
b. 29 is a prime number.
Homework 2.1
Copyright © 2014 – Luis Soto-Ortiz 115
c. 2 is a composite number.
d. 777 is a prime number.
e. 11 is a prime number.
f. 35 is a prime number.
g. 450 is a composite number.
h. 3 is a prime number.
i. 21 is a composite number.
j. 3700 is a composite number.
k. 144 is a prime number.
l. 27 is a prime number.
12. Write a table of all the factors of 100.
13. True or false:
a. 27 is a prime number.
b. 63 is a composite number.
c. 47 is a composite number.
d. 33 is a prime number.
e. 286 is a composite number.
f. 121 is a prime number.
g. 360 is a prime number.
h. 8,674 is a prime number.
i. 835 is a composite number.
Copyright © 2014 – Luis Soto-Ortiz 116
j. 23 is a composite number.
k. 2 is a prime number.
l. 999 is a prime number.
14. T/F _____ 63 is divisible by 7.
15. T/F _____ 532 is divisible by 3.
16. T/F _____ 2,763 is divisible by 9.
17. T/F _____ 322 is divisible by 4.
18. T/F _____ 32 is divisible by 8.
19. T/F _____ 430 is divisible by 10.
20. T/F _____ 72 is divisible by 5.
Copyright © 2014 – Luis Soto-Ortiz 117
9 325 325 9 .
12 240 12 × 20 = 240 ( 240 12 ).
HW 2.1 Solutions:
1.
2.
3. A. True B. True C. True D. True E. True F. False G. False
4. Yes 5. No 6. Yes 7. Yes 8. No
9. 10.
11. a. False b. True c. False d. False e. True f. False g. True h. True i. True
j. True k. False l. False
12.
13. a. False b. True c. False d. False e. True f. False g. False h. False i. True j. False
k. True l. False
14. T 15. F 16. T 17. F 18. T 19. T 20. F
44
1 44
2 22
4 11
150
1 150
2 75
3 50
5 30
6 25
10 15
Answer:
100
1 100
2 50
4 25
5 20
10 10
Copyright © 2014 – Luis Soto-Ortiz 118
Section 2.2 – Rules of Divisibility
In the previous section, we learned what it means for a number to be
divisible by another number. In particular, if we have × = , this means that
and are both factors of . This also means if we divide by or by , the
remainder will be zero. Therefore, × = means that is divisible by and
that is divisible by . For example, since 15 × 2 = 30, both 15 and 2 are factors
of 30. This means that 30 is divisible by 15 and that 30 is divisible by 2. This means
that we get a zero remainder when we divide 30 by 2 and a zero reminder when
we divide 30 by 15. Checking that the remainder is zero is a way to test
divisibility.
Unfortunately, the long division process might be time consuming in some
instances, depending on the numbers that are being divided. Therefore, it is
advantageous to memorize the following divisibility rules of whole numbers and
apply them as appropriate. There are many divisibility rules, but only the most
basic and easy to remember are presented in this table.
Divisible
by Condition Examples
1 All whole numbers are divisible by 1. 0,1,2,3,4,5,6,7,8,9,10,11,12,13,…
2 If the number is even. 0,2,4,6,8,10,12,14,16,18,20,22,…
3 If the sum of the digits is divisible by 3. 0,3,6,9,12,15,18,21,24,27,30,33,…
4 If the 2 rightmost digits are divisible by 4. 0,4,8,12,16,20,24,28,32,36,40,44,…
5 If the number ends with a 5 or 0. 0,5,10,15,20,25,30,35,40,45,50,...
6 If the number is divisible by 2 and by 3. 0,6,12,18,24,30,36,42,48,54,60,…
9 If the sum of the digits is divisible by 9. 0,9,18,27,36,45,54,63,72,81,90,…
10 If the number ends with 0 0,10,20,30,40,50,60,70,80,90,100,…
2
1 5 3 0
- 3 0
0
1 5
2 3 0
- 2
1 0
1 0
0
F
F M - :
0 remainder
factors
multiple
factors
multiple
Copyright © 2014 – Luis Soto-Ortiz 119
Determine whether the number 345,726 is divisible by 1, 2, 3, 5 or 6.
Determine whether the number 68,970 is divisible by 1, 2, 3, 4, 9 or 10.
Example 2.2.1
Answer:
345,726 is divisible by 1 because all whole numbers are divisible by 1.
345,726 is divisible by 2 because 345,726 is an even number.
345,726 is divisible by 3 because the sum of the digits 3+4+5+7+2+6 = 27 and 27
is divisible by 3.
345,726 is not divisible by 5 because the rightmost digit is not 5 or 0.
345,726 is divisible by 6 because 345,726 is divisible by 2 and by 3.
Example 2.2.2
Answer:
68,970 is divisible by 1 because all whole numbers are divisible by 1.
68,970 is divisible by 2 because 68,970 is an even number.
68,970 is divisible by 3 because the sum of the digits 6+8+9+7= 30 and 30 is
divisible by 3.
68,970 is not divisible by 4 because the number formed by the 2 rightmost
digits is 70, but 70 is not divisible by 4.
68,970 is not divisible by 9 because the sum of the digits 6+8+9+7= 30 and 30 is
not divisible by 9.
68,970 is divisible by 10 because the rightmost digit is a zero.
Copyright © 2014 – Luis Soto-Ortiz 120
Determine whether 476,306 is divisible by 9 by applying an appropriate divisibility
rule. Check your answer by performing the long division.
Answer: Since the sum of the digits is 4+7+6+3+0+6 = 26 and 26 is not divisible by
9, this means that 476,306 is not divisible by 9 either. To check the answer, we
perform the long division and note that the remainder is not zero, as expected.
Determine whether 128,975 is divisible by 5 by applying an appropriate divisibility
rule. Check your answer by performing the long division.
Answer: Since the number 128,975 has a digit 5 in the ones place, the number
128,975 is divisible by 5. To check the answer, we perform the long division and
note that the remainder is zero, as expected.
5 2 9 2 2
9 4 7 6 3 0 6
- 4 5
2 6
- 1 8
8 3
- 8 1
2 0
- 1 8
2 6
1 8
8
F
F M
- :
0
Example 2.2.3
Example 2.2.4
Since the reminder is not zero, 9 is not
a factor of 476,306. We also conclude
that 476,306 is not a multiple of 9.
Copyright © 2014 – Luis Soto-Ortiz 121
Instructional videos on the application of the Rules of Divisibility can be found in
the following websites:
http://www.youtube.com/watch?v=AXlz_dHmye4
http://www.youtube.com/watch?v=kBhbv4AVDlI
The following questions ask you to determine whether a number is a factor of the
given number. You may use any method to determine this, including the rules of
divisibility that were presented in this section.
1. Is 2 a factor of 7,986?
2. Is 8 a factor of 6039?
3. Is 5 a factor of 34,780?
4. Is 8 a factor of 7,432?
5. Is 10 a factor of 7,901?
2 5 7 9 5
5 1 2 8 9 7 5
- 1 0
2 8
- 2 5
3 9
- 3 5
4 7
- 4 5
2 5
2 5
0
F
F M
- :
0
Classwork 2.2
A zero remainder means that the
number 128,975 is divisible by 5.
Hence, 5 is a factor of 128,975 and
128,975 is a multiple of 5.
Copyright © 2014 – Luis Soto-Ortiz 122
6. Is 7 a factor of 7,910?
7. Is 9 a factor of 666?
8. Is 538 divisible by 2?
9. Is 7,872 divisible by 3?
10. Is 345 divisible by 5?
11. Is 9 a factor of 3,673,909?
12. Is 4 a factor of 845,912?
13. Is 2 a factor of 67,932,663?
14. Is 3 a factor of 852,504?
15. Is 9 a factor of 852,504?
16. Is 10 a factor of 89,015?
17. Is 9 a factor of 10,203?
18. Is 6,340 divisible by 5?
19. Is 48 divisible by 3?
20. Is 48 divisible by 2?
21. Is 8,360 divisible by 4?
22. Is 34,785 divisible by 4?
23. Is 678,021 divisible by 5?
24. Is 678,021 divisible by 3?
25. Is 30 divisible by 10?
26. Is 827 divisible by 2?
27. Is 7,212 divisible by 3?
Copyright © 2014 – Luis Soto-Ortiz 123
28. Is 9,948 divisible by 4?
29. Is 346,915 divisible by 5?
30. Is 6,783 divisible by 9?
Copyright © 2014 – Luis Soto-Ortiz 124
CW 2.2 Solutions:
1. Yes 2. No 3. Yes 4. Yes 5. No 6. Yes 7. Yes 8. Yes 9. Yes 10. Yes
11. No 12. Yes 13. No 14. Yes 15. No 16. No 17. No 18. Yes
19. Yes 20. Yes 21. Yes 22. No 23. No 24. Yes 25. Yes 26. No
27. Yes 28. Yes 29. Yes 30. No
Copyright © 2014 – Luis Soto-Ortiz 125
The following questions ask you to determine whether a number is a factor of the
given number. You may use any method to determine this, including the rules of
divisibility that were presented in this section.
1. Is 9 a factor of 504?
2. Is 6 a factor of 530?
3. Is 2 a factor of 687,421?
4. Is 5 a factor of 120?
5. Is 7 a factor of 821?
6. Is 10 a factor of 16,785?
7. Is 9 a factor of 440?
8. Is 470 divisible by 2?
9. Is 16,002 divisible by 3?
10. Is 120 divisible by 3?
11. Is 4 a factor of 32,719?
12. Is 2 a factor of 97,456,031?
13. Is 6 a factor of 34,692?
14. Is 3 a factor of 600?
15. Is 9 a factor of 5,555?
16. Is 2 a factor of 90?
17. Is 3 a factor of 90?
18. Is 80 divisible by 3?
Homework 2.2
Copyright © 2014 – Luis Soto-Ortiz 126
19. Is 145 divisible by 5?
20. Is 620 divisible by 4?
21. Is 9,879 divisible by 9?
22. Is 10 divisible by 2?
23. Is 774,645 divisible by 5?
24. Is 666,666 divisible by 9?
25. Is 5,145 divisible by 10?
26. Is 654 divisible by 2?
27. Is 16,428 divisible by 4?
28. Is 1,736 divisible by 3?
29. Is 417,370 divisible by 5?
30. Is 720 divisible by 4?
Copyright © 2014 – Luis Soto-Ortiz 127
HW 2.2 Solutions:
1. Yes 2. No 3. No 4. Yes 5. No 6. No 7. No 8. Yes 9. Yes 10. Yes
11. No 12. No 13. Yes 14. Yes 15. No 16. Yes 17. Yes 18. No
19. Yes 20. Yes 21. No 22. Yes 23. Yes 24. Yes 25. No 26. Yes
27. Yes 28. No 29. Yes 30. Yes
Copyright © 2014 – Luis Soto-Ortiz 128
Section 2.3 – Prime Factorization
Recall that a prime number has exactly two different factors: 1 and the
number itself. For your convenience, here again is a list of all the prime numbers
that are less than 1000:
Table of Prime Numbers Less Than 1,000
2 3 5 7 11 13 17 19 23
29 31 37 41 43 47 53 59 61 67
71 73 79 83 89 97 101 103 107 109
113 127 131 137 139 149 151 157 163 167
173 179 181 191 193 197 199 211 223 227
229 233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337 347
349 353 359 367 373 379 383 389 397 401
409 419 421 431 433 439 443 449 457 461
463 467 479 487 491 499 503 509 521 523
541 547 557 563 569 571 577 587 593 599
601 607 613 617 619 631 641 643 647 653
659 661 673 677 683 691 701 709 719 727
733 739 743 751 757 761 769 773 787 797
809 811 821 823 827 829 839 853 857 859
863 877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997
We have learned that whole numbers have factors, and thus can be written in
factorized form. For example, some factorizations of the number 360 are
360 = 6 × 5 × 12
360 = 1 × 9 × 4 × 10
360 = 18 × 5 × 4
360 = 1 × 360
These are four different factorizations
of 360 because when we multiply the
whole numbers the product is 360.
Copyright © 2014 – Luis Soto-Ortiz 129
A factorization of a number shows factors that multiplied together give the
original number. Note that although 4, 6 and 5 are factors of 360, the expression
4 × 6 × 5 is not a factorization of 360 because 4 × 6 × 5 360. Recall that the
symbol means “not equal to.”
Write 5 different factorizations of 2,000.
In some applications, it is useful to factorize a whole number using only prime
factors. The prime factorization of a number entails “breaking” or “splitting” a
number into factors that are prime numbers, and that gives back the original
number when we multiply the prime factors.
For example, the prime factorization of 2,000 is 2 × 2 × 2 × 2 × 5 × 5 × 5
because all these factors of 2,000 are prime numbers, and when we multiply
2 × 2 × 2 × 2 × 5 × 5 × 5 we get back 2,000.
The prime factorization of 90 is 2 × 3 × 3 × 5 because 2 × 3 × 3 × 5 = 90 and
the numbers 2, 3 and 5 are prime.
Note: 1 is not a prime number because it has only one factor: itself 1 × 1 = 1 .
Example 2.3.1
Answer:
2,000 = 1 × 100 × 20
2,000 = 10 × 4 × 50
2,000 = 1 × 2 × 4 × 5 × 10 × 5
2,000 = 1 × 2000
2,000 = 50 × 40
Copyright © 2014 – Luis Soto-Ortiz 130
×
×
×
×
A method to find the prime factorization of any whole number involves
constructing a tree of factors. Each factor appearing in the tree must be either a
prime number or a composite number. Hence, 1 should not appear in a tree of
factors because 1 is neither prime nor composite. The approach to construct a
tree of factors is to split, or factor, the original number into a product of prime
and/or composite factors, and then continue splitting these factors until we are
left with prime factors at the end of the branches.
For example, to find the prime factorization of 12, we begin by factoring 12 in any
way we choose, as long as the factors are prime or composite. At the end of the
branches, we will be left with only prime numbers that multiplied together give
the original number we started with (12).
The numbers in red are the prime factors of 12, and so the prime factorization of
12 in expanded form is × × . If you are familiar with exponents, you can
write the prime factorization in exponential form as × . You will learn more
about exponential notation in Section 2.5.
Write the prime factorization of 45 in expanded form.
Using exponents, the prime factorization of 45 in exponential form is given by
45 = 5 × 3 .
Example 2.3.2
Answer: 45 = 5 × 3 × 3
Copyright © 2014 – Luis Soto-Ortiz 131
×
× ×
×
×
× ×
Write the prime factorization of 120 in expanded form.
Using exponents, the prime factorization of 120 is 120 = 5 × 2 × 3 .
Write the prime factorization of 350 in expanded form.
In exponential form, the answer is 350 = 2 × 5 × 7 .
Example 2.3.3
Answer: 120 = 5 × 2 × 2 × 2 × 3
Example 2.3.4
Answer: 350 = 2 × 5 × 5 × 7
Copyright © 2014 – Luis Soto-Ortiz 132
Write the prime factorization of 504 in expanded form.
In exponential form, the answer is 504 = 2 × 3 × 7
Instructional video on finding the prime factorization of a whole number:
http://www.youtube.com/watch?v=YKXE2rMKPYA
The following website has an interactive tool to help you construct a tree of
factors to find the prime factorization of any whole number:
http://www.softschools.com/math/factors/factor_tree/
Write the prime factorization of each number.
1. 70
2. 100
3. 231
Example 2.3.5
Answer: 504 = 2 × 2 × 2 × 3 × 3 × 7
×
×
×
×
×
Classwork 2.3
Copyright © 2014 – Luis Soto-Ortiz 133
4. 441
5. 420
6. 800
7. 3,600
8. 26
9. 98
10. 1,000
11. 111
12. 666
13. 385
14. 900
15. 64
16. 125
17. 4,000
18. 9
19. 52
20. 350
21. 1,600
22. 36
23. 280
24. 243
25. 625
Copyright © 2014 – Luis Soto-Ortiz 135
CW 2.3 Solutions:
1. 70 = 2 × 5 × 7
2. 100 = 2 × 2 × 5 × 5
3. 231 = 3 × 7 × 11
4. 441 = 3 × 3 × 7 × 7
5. 420 = 2 × 2 × 3 × 5 × 7
6. 800 = 2 × 2 × 2 × 2 × 2 × 5 × 5
7. 3,600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5
8. 26 = 2 × 13
9. 98 = 2 × 7 × 7
10. 1,000 = 2 × 2 × 2 × 5 × 5 × 5
11. 111 = 3 × 37
12. 666 = 2 × 3 × 3 × 37
13. 385 = 5 × 7 × 11
14. 900 = 2 × 2 × 3 × 3 × 5 × 5
15. 64 = 2 × 2 × 2 × 2 × 2 × 2
16. 125 = 5 × 5 × 5
17. 4,000 = 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5
18. 9 = 3 × 3
19. 52 = 2 × 2 × 13
20. 350 = 2 × 5 × 5 × 7
21. 1,600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
22. 36 = 2 × 2 × 3 × 3
23. 280 = 2 × 2 × 2 × 5 × 7
24. 243 = 3 × 3 × 3 × 3 × 3
25. 625 = 5 × 5 × 5 × 5
26. 726 = 2 × 3 × 11 × 11
27. 2,940 = 2 × 2 × 3 × 5 × 7 × 7
28. 570 = 2 × 3 × 5 × 19
29. 3,465 = 3 × 3 × 5 × 7 × 11
30. 936 = 2 × 2 × 2 × 3 × 3 × 13
Copyright © 2014 – Luis Soto-Ortiz 136
Write the prime factorization of each number.
1. 735
2. 180
3. 924
4. 60
5. 2,300
6. 64
7. 80
8. 4,620
9. 81
10. 690
11. 6
12. 700
13. 57
14. 582
15. 105
16. 40
17. 225
18. 9,600
19. 144
20. 72
Homework 2.3
Copyright © 2014 – Luis Soto-Ortiz 137
21. 8
22. 7
23. 2,187
24. 1,750
25. 372
26. 4,455
27. 205
28. 9,936
29. 1,100
30. 85
Copyright © 2014 – Luis Soto-Ortiz 138
HW 2.3 Solutions:
1. 735 = 3 × 5 × 7 × 7
2. 180 = 2 × 2 × 3 × 3 × 5
3. 924 = 2 × 2 × 3 × 7 × 11
4. 60 = 2 × 2 × 3 × 5
5. 2,300 = 2 × 2 × 5 × 5 × 23
6. 64 = 2 × 2 × 2 × 2 × 2 × 2
7. 80 = 2 × 2 × 2 × 2 × 5
8. 4,620 = 2 × 2 × 3 × 5 × 7 × 11
9. 81 = 3 × 3 × 3 × 3
10. 690 = 2 × 3 × 5 × 23
11. 6 = 2 × 3
12. 700 = 2 × 2 × 5 × 5 × 7
13. 57 = 3 × 19
14. 582 = 2 × 3 × 97
15. 105 = 3 × 5 × 7
16. 40 = 2 × 2 × 2 × 5
17. 225 = 3 × 3 × 5 × 5
18. 9,600 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 5
19. 144 = 2 × 2 × 2 × 2 × 3 × 3
20. 72 = 2 × 2 × 2 × 3 × 3
21. 8 = 2 × 2 × 2
22. 7 = 7 a prime number is its own prime factorization
23. 2,187 = 3 × 3 × 3 × 3 × 3 × 3 × 3
24. 1,750 = 2 × 5 × 5 × 5 × 7
25. 372 = 2 × 2 × 3 × 31
26. 4,455 = 3 × 3 × 3 × 3 × 5 × 11
27. 205 = 5 × 41
28. 9,936 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 23
29. 1,100 = 2 × 2 × 5 × 5 × 11
30. 85 = 5 × 17