Factors and Multiples

Multiples of a Number

Multiples of a number are related to factors of a number. A multiple of a number is that number’s product with some integer. For example, if a hardware store sells a type of screw that only comes in packs of 20, customers must buy these screws in multiples of 20: that is, 20, 40, 60, 80, and so on. (Technically, 0 is also a multiple.) These numbers are equal to 20 × 1, 20 × 2, 20 × 3, 20 × 4, and so on. Similarly, measurements in feet represent multiples of 12 inches. A (whole-number) measurement in feet would be equivalent to 12 inches, 24 inches, 36 inches, and so on.

When counting by twos or threes, multiples are used. But because the multiples of a number are the product of that number with the integers, multiples can also be negative. For the number 2, the multiples are the set {…, –6, –4, –2, 0, 2, 4, 6,…}, where the ellipsis dots indicate that the set continues the pattern indefinitely in both directions. Also, the number can be any real number: the multiples of π (approximately 3.14) are {…, –3π, –2π, –1π, 0, 1π, 2π, 3π,…}. Note that the notation 2π, for example, means 2 × π.

The positive multiples (along with 0) of a whole number are all numbers for which that whole number is a factor. For instance, the positive multiples of 5 are 0, 5, 10, 15, 20, 25, 30, and so on. That full set contains all (whole) numbers for which 5 is a factor. Thus, one number is a multiple of a second number if the second number is a factor of the first.

Example

Prime and Composite Numbers

For some real-world applications, such as cryptography, factors and multiples play an important role. One important way to classify whole numbers is by whether they are prime or composite. A prime number is any whole (or natural) number greater than 1 that has only itself and 1 as factors. The smallest example is 2: because 2 only has 1 and 2 as factors, it is prime. Composite numbers have at least one factor other than 1 and themselves. The smallest composite number is 4: in addition to 1 and 4, it has 2 as a factor.

Determining whether a number is prime can be extremely difficult—hence its value in cryptography. One simple test that works for some numbers is to check whether the number is even or odd. An even number is divisible by 2; an odd number is not. To determine whether a number is even or odd, look at the last (rightmost) digit. If that digit is even (0, 2, 4, 6, or 8), the number is even. Otherwise, it is odd. Another simple test works for multiples of 3. Add all the digits in the number. If the sum is divisible by 3, the original number is also divisible by 3. This rule can be successively applied multiple times until the sum of digits is manageable. That number is then composite.

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Be Careful!

Avoid the temptation to call 1 a prime number. Although it only has itself and 1 as factors, those factors are the same number. Hence, 1 is fundamentally different from the prime numbers, which start at 2.

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Example

Prime Factorization

Determining whether a number is prime, even for relatively small numbers (less than 100), can be difficult. One tool that can help both solve this problem and identify all factors of a number is prime factorization. One way to do prime factorization is to make a factor tree.

The procedure below demonstrates the process.

Step by Step

1. Write the number you to factor
2. If the number is prime, stop. Otherwise, go to Step 3.
3. Find any two factors of the number and write them on the line below the number.
4. “Connect” the factors and the number using line segments. The result will look somewhat like an inverted tree, particularly as the process continues.
5. Repeat Steps 2–4 for all composite factors in the tree.

The numbers in the factor tree are either “branches” (if they are connected downward to other numbers) or “leaves” (if they have no further downward connections). The leaves constitute all the prime factors of the original number: when multiplied together, their product is that number. Moreover, any product of two or more of the leaves is a factor of the original number. Thus, using prime factorization helps find any and all factors of a number, although the process can be tedious when performed by hand (particularly for large numbers). Below is a factor tree for the number 96. All the leaves are circled for emphasis.

Example

Let’s Review

• A whole number is divisible by all of its factors, which are also whole numbers by definition.
• Multiples of a number are all possible products of that number and the integers.
• A prime number is a whole number greater than 1 that has no factors other than itself and 1.
• A composite number is a whole number greater than 1 that is not prime (that is, it has factors other than itself and 1).
• Even numbers are divisible by 2; odd numbers are not.
• Prime factorization yields all the prime factors of a number. The factor-tree method is one way to determine prime factorization.