2.02
Factoring Numbers
from Basic Algebra: One Step at
a Time © 2002
P.
133-138
Dr. Robert J. Rapalje
Seminole Community College
Sanford, FL 32773
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE
END OF THIS PAGE
It is
frequently useful to write numbers in a factored form. For
example, if you were asked to factor the number 15, what would you say?
You probably would instinctively reply 3 times 5 (or 5 times 3). The
numbers 3 and 5 are factors of 15, since 3 and 5 divide evenly into 15.
That is to say, if you divide 15 by 3 or by 5, there is no remainder.
Notice that 1 and 15 are also factors of 15, since 1 and 15 also divide
evenly into 15. Likewise, the number 26 can be written as 2
@
13, so 2 and 13 (also 1 and 26) are factors of 26. Since the number 91
can be written as 7 @
13, 7 and 13 (also 1 and 91) are factors of 91.
Question: How would you factor 18? In
this case, there is more than one possibility. You could say 18 is 6
times 3, or you could say 2 times 9. If you continue to break the numbers
down, you will get:
18 = 6 @
3 18 = 2
@
9
= 2 @
3 @
3 = 2
@
3 @
3
= 2 @
32
= 2 @
32
The
factors of 18 are 1, 2, 3, 6, 9, and 18.
Each of
the numbers 15, 18, 26, and 91 could be written as the product of smaller
numbers. What if you were asked to factor 17? What about 37? You will
not be able to find smaller numbers that can be used to break down the 17
or the 37, as you did with 15, 18, 26, and 91. Since the numbers 17 and
37 cannot be expressed as the product of smaller numbers, they are called
prime numbers. A prime number is any number
larger than 1 that has exactly two factors: 1 and itself. A
composite number is any number that has more than two factors.
Composite numbers may be broken down into the product of smaller numbers.
The number 1 is a special number in that it is neither prime nor
composite.
DEFINITIONS
A prime number is a number larger
than 1 that cannot be expressed as the product of two
smaller numbers.
A composite number is a number
that can be expressed as the product of two smaller numbers.
The number one (1) is neither prime nor
composite.
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Complete the following list of prime
numbers from 2 to 101.
2, 3, ___, ___, ___, ___, ___, 19, ___, ___,
___, ___, 41, ___, ___, ___, 59,
___, ___, 71, ___, ___, ___, 89, ___,
101.
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Perhaps
you noticed that the larger the numbers, the harder it its to tell whether
or not the numbers are prime. At this point, it will be helpful to learn
a few Ashortcuts@
to determine divisibility by certain numbers. With or without these
shortcuts, a calculator will be very helpful.
1.
Divisibility using a calculator. To determine if a number is
divisible by a second number with a calculator, just divide the first
number by the second number to see if it comes out a whole number on the
calculator. For example, to see if 5289 is divisible by 41, type [5289] [)]
[41] [=]. You will see that the answer is the whole number 129. Is 5289
divisible by 73? When you divide, the answer is NOT a whole number, but
rather it comes out to a decimal (like 72.45205 and more!). Thus you can
see that 5289 is NOT divisible by 73.
2. Divisibility
by 2. What numbers are divisible by 2? You probably already know the
answer‑‑Aeven@
numbers, numbers
whose last digit is even.
3.
Divisibility by 5. What numbers are divisible by 5? Again, you
probably already know the answer--numbers that end in a 5 or a 0.
4.
Divisibility by 10. What numbers are divisible by 10? Again, you
already know this--numbers that end in a 0.
5.
Divisibility by 3. What numbers are divisible by 3? Look at some
numbers that you know are divisible by 3, like 15, 18, 33, 42, 66, 72, and
75 to name a few. Notice that in every case, the sum of the digits is
also divisible by 3. On the other hand, try some numbers like 13, 26, 41,
43, 44, etc. that are not divisible by 3. Notice that the sum of
the digits also is not divisible by 3.
6.
Divisibility by 9. Look at some numbers that you know are divisible
by 9: 18, 27, 36, 45, 54, 63, 72, 81, and 90. Notice that in every case,
the sum of the digits is also divisible by 9. On the other hand, try some
numbers like 12, 26, 48, 53, 84, etc. that are not divisible by 9.
Notice that the sum of the digits also is not divisible by 9.
DIVISIBILITY BY 3 AND 9
If a number is divisible by 3 or by 9,
then the sum of its digits is also divisible by 3 or by 9, and
vice-versa.
If the sum of the digits is not
divisible by 3 or by 9, then the number is not divisible by
3 or by 9, and vice-versa.
NOTICE THAT THIS RULE ONLY WORKS FOR 3
AND 9.
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7.
Divisibility by 4. If the last two digits are divisible by 4, then
the number is divisible by 4.
8.
Divisibility by 6. To be divisible by 6, a number must be divisible
by 2 and also by 3. In other words, it must be an even number that is
divisible by 3.
You
probably noticed that for larger numbers it becomes more difficult to
determine if the number is prime. For very large numbers it becomes very
difficult. As a general rule, to determine if a number is prime, you
must check to see if any prime numbers up to and including the square root
of the number divide into it evenly. For example, to determine if 97 is a
prime number, first use your calculator to find the square root of 97,
which is 9.8. Then check to see if 97 is divisible by any prime number up
to 9.8--that is, 2,3,5, or 7.
The
following examples and exercises show how to give the prime factorization
of a number. These are called
Afactor
trees.@
The idea is to find
any number that divides into the number, then use the calculator to
divide and find the other number. Keep breaking down the numbers until
you get prime numbers. Circle the prime numbers. The product of these
circled numbers is the prime factorization of the given number. The
examples show the process.
Express the numbers as a product of prime numbers:
EXAMPLE 1. 72
EXAMPLE 2. 750
EXAMPLE 3. 29700
Solutions:
72
750
29700
9 8
75 10
297 100
3 3 2 4 3 25
2 5
9 33 10 10
2 2
5 5
3 3 3 11 2 5 2 5
ANSWERS: 72 = 23
@
32
750 = 2
@
3
@
53
29700 = 22
@
33
@
52
@
11
EXERCISES. In each of the
following, express the numbers as the product of prime numbers.
1.
14
2. 33
3. 35
4. 77
5. 19
6. 37
7. 25
8. 49
9. 93
10. 87
11. 51
12. 94
13.
70
14. 28
15. 45
16.
54
17. 99
18. 105
19.
60
20. 80
21. 135
22.
700 23. 760
24. 790
25.
128 26.
450
27. 2000
28.
8400 29.
3060
30. 13,320
31. 10,000
32. 100,000
33. 52,200
34.
20,025
35. 55,800
36. 25,600
EXTRA CHALLENGE: Factor each of the following
into primes (a calculator may help).
37.
1,000,000
38. 1,000,000,000
39.
111,111
40. 111,111,111
[Hint: 333667 is a prime number!]
ANSWERS
2.02
p.134:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101.
p. 136-145:
1.
2 7; 2. 3 11; 3. 5 7; 4.
7 11; 5. Prime; 6. Prime; 7. 52;
8. 72; 9. 3
31; 10. 3 29; 11. 3 17;
12. 2 47; 13. 2 5
7; 14. 22 7; 15.
32 5; 16. 2 33;
17. 32 11; 18.
3 5 7; 19. 22 3
5; 20. 24 5;
21.
33
5; 22. 22 52
7; 23. 23 5 19;
24. 2 5 79; 25. 27;
26. 2 32 52;
27. 24 53;
28. 24 3 52
7;
29. 22
32 5 17; 30. 23
32 5 37; 31. 24
54; 32. 25
55; 33. 23
32 52
29; 34. 32 52
89;
35. 23
32 52
31; 36. 210 52;
37. 26 56;
38. 29 59;
39. 3 7 11 13 37; 40. 32
37 333667.
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