Factors and Multiples
In the problem 3 x 4 = 12, 3 and 4 are factors and 12 is the product.
Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).
The number 12 is a multiple of 3, because it can be divided evenly by 3.
3 x 4 = 12
3 and 4 are both factors of 12
12 is a multiple of both 3 and 4.
A factor is simply a number that is multiplied to get a product. Factoring a number means taking the number apart to find its factors; it's like multiplying in reverse. Here are lists of all the factors of 16, 20, and 45.
16 --> 1, 2, 4, 8, 16
20 --> 1, 2, 4, 5, 10, 20
45 --> 1, 3, 5, 9, 15, 45
12 --> 12, 24, 36, 48, 60, . . .
5 --> 5, 10, 15, 20, 25, . . .
7 --> 7, 14, 21, 28, 35, . . .
Factors are either composite numbers or prime numbers. A prime number has only two factors, one and itself, so it cannot be divided evenly by any other numbers. Here's a list of prime numbers up to 100. You can see that none of these numbers can be factored any further.
PRIME NUMBERS to 100
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
A composite number is any number that has more than two factors. Here's a list of composite numbers up to 20. You can see that they can all be factored further. For example, 4 equals 2 times 2, 6 equals 3 times 2, 8 equals 4 times 2, and so forth.
By the way, zero and one are considered neither prime nor composite numbers-they're in a class by themselves!
COMPOSITE NUMBERS up to 20
4,6,8,9,10,12,14,15,16,18,20
You can write any composite number as a product of prime factors. This is called prime factorization. To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number. Let's use this method to find the prime factors of 168. Since 168 is even, we start by dividing it by the smallest prime number, 2. 168 divided by 2 is 84.
84 divided by 2 is 42. 42 divided by 2 is 21. Since 21 is not divisible by 2, we try dividing by 3, the next biggest prime number. We find that 21 divided by 3 equals 7, and 7 is a prime number. We know 168 is now fully factored. We simply list the divisors to write the factors of 168.
168 ÷ 2 = 84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7 Prime number
prime factors = 2 × 2 × 2 × 3 × 7
To check the answer, multiply these factors and make sure they equal 168.
Here are the prime factors of the composite numbers between 1 and 20.
4 = 2 × 2
6 = 3 × 2
8 = 2 × 2 × 2
9 = 3 × 3
10 = 5 × 2
12 = 3 × 2 × 2
14 = 7 × 2
15 = 5 × 3
16 = 2 × 2 × 2 × 2
18 = 3 × 3 × 2
20 = 5 × 2 × 2
The greatest common factor, or GCF, is the greatest factor that divides two numbers. To find the GCF of two numbers:
GCF
It's often useful in solving math problems to be able to find the largest factor that divides two numbers. We call this the greatest common factor, or GCF. Let's find the GCF of 30 and 45. First we find the prime factors of each number, using prime factorization.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Next, identify those prime factors that both numbers have in common, and multiply them. Here, both 3 and 5 are common factors. The GCF is 3 times 5, or 15.
3 × 5 = 15 <— GCF
EXAMPLES
Find the GCF of these pairs of numbers.
14, 49
Solution: List the prime factors of each.
14: 2 × 7
49: 7 × 7
7 is the only common factor; therefore, 7 is the GCF.
15, 75
Solution: List the prime factors of each.
15: 3 × 5
75: 3 × 5 × 5
3 and 5 are common; therefore, 3 × 5 = 15 is the GCF.
LCM
A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24, ....
The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.
The least common multiple, or LCM, is another number that's useful in solving many math problems. Let's find the LCM of 30 and 45. One way to find the least common multiple of two numbers is to first list the prime factors of each number.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
2: one occurrence
3: two occurrences
5: one occurrence
2 × 3 × 3 × 5 = 90 <— LCM
After you've calculated a least common multiple, always check to be sure your answer can be divided evenly by both numbers.
EXAMPLES
Find the LCM of these sets of numbers.
3, 9, 21
Solution: List the prime factors of each.
3: 3
9: 3 × 3
21: 3 × 7
Multiply each factor the greatest number of times it occurs in any of the numbers. 9 has two 3s, and 21 has one 7, so we multiply 3 two times, and 7 once. This gives us 63, the smallest number that can be divided evenly by 3, 9, and 21. We check our work by verifying that 63 can be divided evenly by 3, 9, and 21.
12, 80
Solution: List the prime factors of each.
12: 2 × 2 × 3
80: 2 × 2 × 2 × 2 × 5 = 80
Multiply each factor the greatest number of times it occurs in either number. 12 has one 3, and 80 has four 2's and one 5, so we multiply 2 four times, 3 once, and five once. This gives us 240, the smallest number that can be divided by both 12 and 80. We check our work by verifying that 240 can be divided by both 12 and 80.
In the problem 3 x 4 = 12, 3 and 4 are factors and 12 is the product.
Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).
The number 12 is a multiple of 3, because it can be divided evenly by 3.
3 x 4 = 12
3 and 4 are both factors of 12
12 is a multiple of both 3 and 4.
A factor is simply a number that is multiplied to get a product. Factoring a number means taking the number apart to find its factors; it's like multiplying in reverse. Here are lists of all the factors of 16, 20, and 45.
16 --> 1, 2, 4, 8, 16
20 --> 1, 2, 4, 5, 10, 20
45 --> 1, 3, 5, 9, 15, 45
12 --> 12, 24, 36, 48, 60, . . .
5 --> 5, 10, 15, 20, 25, . . .
7 --> 7, 14, 21, 28, 35, . . .
Factors are either composite numbers or prime numbers. A prime number has only two factors, one and itself, so it cannot be divided evenly by any other numbers. Here's a list of prime numbers up to 100. You can see that none of these numbers can be factored any further.
PRIME NUMBERS to 100
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
A composite number is any number that has more than two factors. Here's a list of composite numbers up to 20. You can see that they can all be factored further. For example, 4 equals 2 times 2, 6 equals 3 times 2, 8 equals 4 times 2, and so forth.
By the way, zero and one are considered neither prime nor composite numbers-they're in a class by themselves!
COMPOSITE NUMBERS up to 20
4,6,8,9,10,12,14,15,16,18,20
You can write any composite number as a product of prime factors. This is called prime factorization. To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number. Let's use this method to find the prime factors of 168. Since 168 is even, we start by dividing it by the smallest prime number, 2. 168 divided by 2 is 84.
84 divided by 2 is 42. 42 divided by 2 is 21. Since 21 is not divisible by 2, we try dividing by 3, the next biggest prime number. We find that 21 divided by 3 equals 7, and 7 is a prime number. We know 168 is now fully factored. We simply list the divisors to write the factors of 168.
168 ÷ 2 = 84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7 Prime number
prime factors = 2 × 2 × 2 × 3 × 7
To check the answer, multiply these factors and make sure they equal 168.
Here are the prime factors of the composite numbers between 1 and 20.
4 = 2 × 2
6 = 3 × 2
8 = 2 × 2 × 2
9 = 3 × 3
10 = 5 × 2
12 = 3 × 2 × 2
14 = 7 × 2
15 = 5 × 3
16 = 2 × 2 × 2 × 2
18 = 3 × 3 × 2
20 = 5 × 2 × 2
The greatest common factor, or GCF, is the greatest factor that divides two numbers. To find the GCF of two numbers:
- List the prime factors of each number.
- Multiply those factors both numbers have in common. If there are no common prime factors, the GCF is 1.
GCF
It's often useful in solving math problems to be able to find the largest factor that divides two numbers. We call this the greatest common factor, or GCF. Let's find the GCF of 30 and 45. First we find the prime factors of each number, using prime factorization.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Next, identify those prime factors that both numbers have in common, and multiply them. Here, both 3 and 5 are common factors. The GCF is 3 times 5, or 15.
3 × 5 = 15 <— GCF
EXAMPLES
Find the GCF of these pairs of numbers.
14, 49
Solution: List the prime factors of each.
14: 2 × 7
49: 7 × 7
7 is the only common factor; therefore, 7 is the GCF.
15, 75
Solution: List the prime factors of each.
15: 3 × 5
75: 3 × 5 × 5
3 and 5 are common; therefore, 3 × 5 = 15 is the GCF.
LCM
A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24, ....
The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.
The least common multiple, or LCM, is another number that's useful in solving many math problems. Let's find the LCM of 30 and 45. One way to find the least common multiple of two numbers is to first list the prime factors of each number.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
2: one occurrence
3: two occurrences
5: one occurrence
2 × 3 × 3 × 5 = 90 <— LCM
After you've calculated a least common multiple, always check to be sure your answer can be divided evenly by both numbers.
EXAMPLES
Find the LCM of these sets of numbers.
3, 9, 21
Solution: List the prime factors of each.
3: 3
9: 3 × 3
21: 3 × 7
Multiply each factor the greatest number of times it occurs in any of the numbers. 9 has two 3s, and 21 has one 7, so we multiply 3 two times, and 7 once. This gives us 63, the smallest number that can be divided evenly by 3, 9, and 21. We check our work by verifying that 63 can be divided evenly by 3, 9, and 21.
12, 80
Solution: List the prime factors of each.
12: 2 × 2 × 3
80: 2 × 2 × 2 × 2 × 5 = 80
Multiply each factor the greatest number of times it occurs in either number. 12 has one 3, and 80 has four 2's and one 5, so we multiply 2 four times, 3 once, and five once. This gives us 240, the smallest number that can be divided by both 12 and 80. We check our work by verifying that 240 can be divided by both 12 and 80.