Factors
A factor is an integer that divides evenly into another number.
example 1
The number 18 has a total of 6 factors: 1, 2, 3, 6, 9, and 18. What is the total number
of factors that the number 130 has?
A.
B.
C.
D.
4
6
8
13
One important thing to understand about factors is that they come in pairs. The
factors for 18 are given above as a list, but come from: 1 × 18, 2 × 9, and 3 × 6.
Multiplying each pair of factors together equals 18.
To find all the factors of 130, you need to find each pair of numbers that can be
multiplied together to get a product of 130. One way to do this is to use the opposite
of multiplication, which is division. Start dividing 130 by 2, 3, 4, etc. to see if they
divide evenly into 130. Your first factors will always be 1 and the number itself
(1 × 130 = 130).
We can stop checking numbers at this point because the next number to test is 13, but
we’ve already found 13 as a factor. Notice how each quotient found got smaller and
smaller (65, then 43.3, then 32.5, etc.) while you tested larger and larger integers (2,
then 3, then 4, etc.). As soon as a quotient is found to be smaller than the number you
are testing, you can stop searching for factors, because at this point you’ll start
repeating factors. (Such 10 and 13, and then 13 and 10).
The factors that were found were: 1 × 130, 2 × 65, 5 × 26, and 10 × 13. Or you can list
them out as 1, 2, 5, 10, 13, 26, 65, and 130.
The answer is C.
Prime Numbers
A prime number is a whole number that has only one pair of factors: the number 1 and
itself. This means a prime number cannot be divided evenly by any other whole numbers.
(A whole number is a nonnegative integer: 0, 1, 2, 3, etc.)
The table below shows the first 100 prime numbers:
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
Composite Numbers
Except for 0 and 1, a whole number is either prime or composite. A composite number
is any whole number, greater than one, that is not prime. This means a composite number
has more than two factors, or in other words, it can be evenly divided by another whole
number besides 1 and itself.
All even numbers, except for 2, are composite numbers. (This also means that 2 is the
only even prime number, as shown in the table above.)
Greatest Common Factors
Two or more numbers can have the same factors, called common factors. For example,
the factors of 20 are 1, 2, 4, 5, 10, and 20 and the factors of 30 are 1, 2, 3, 5, 6, 10, 15,
and 30. The common factors of 20 and 30 are 1, 2, 5, and 10. The largest of the common
factors is called the greatest common factor, which is 10 in this example.
To find the greatest common factor among two or more numbers, write out the prime
factorization of each number using the following procedure:
Divide each number by the smallest prime number by which it can be divided. Start with
2 if the number is even, or 3 if the number is odd, and then go up one prime number at a
time (5, 7, 11, etc.) until you have found all of the prime numbers that can be multiplied
together to create the number you started with.
example 2
Write out the prime factorization of 140 and 72.
Once the prime factorization is written out, any common factors can be easily seen:
For the numbers 140 and 72, they each have prime factors of 2 and another 2, so multiply
the common prime factors together, and a greatest common factor of 4 is found.
example 3
What is the prime factorization of 300?
A.
B.
C.
D.
3 · 102
2 · 52 · 6
22 · 3 · 25
22 · 3 · 52
The prime factorization of 300 can be written out as:
The answer is D.
Name ______________________________
Write out the prime factorization of each number.
1) 30
2) 108
3) 17 5
4 ) 2 10
5) 41 6
6 ) 6 30
7) 87 5
8 ) 2 310
Find the greatest common factor of the numbers given.
9) 20, 50
10) 16, 56
11) 36, 60
12) 54, 81
13) 128, 256
14) 136, 168
15) 28, 32, 36
16) 45, 63, 81