Algebra Know-It-ALL

40 Natural Numbers and Integers

We can’t break the number 3 down into a product of natural numbers other than itself and 1.

How about 33? This can break down into the product of 11 and 3:

33 = 11 × 3

We can’t break 11 down into a product of natural numbers other than itself and 1. Now we

have 99 as a product of “unbreakables”:

99 = 11 × 3 × 3

Whenever you have a natural number expressed as a product of other numbers, those other

numbers are called factors. The process of breaking a number down into a product of other

numbers is called factorization or factoring.

Prime and composite numbers

An “unbreakable” natural number is called a prime number, or simply a prime. It’s a natural

number larger than 1 that can only be factored into a product of itself and 1. Table 3-1 lists

the first 24 prime numbers. If this doesn’t go high enough for you, there are plenty of lists of

primes on the Internet.

Any nonprime natural number can be factored into a product of two or more primes.

The numbers in such a product are called the prime factors, and the whole product is

called a composite number. When you want to find the prime factors of a large natural

number, you can get some help from a calculator that has a square root key. The square

root of a number is a smaller number, not always whole, that gives you the original num-

ber when multiplied by itself.

Here’s how the process goes. First, use a calculator to find the square root of the number

you want to factor. Once you have done that, “chop off ” any nonzero digits that might appear

after the decimal point, so you get a whole number. Then add 1 to that number. Call this new

whole number s. Now divide the original number by all the primes (referring to Table 3-1) less

than or equal to s, starting with the largest prime and working your way down. If you ever get

a whole-number quotient as you go through this process, then you know that the divisor and

Table 3-1. The first 24 prime numbers. The number 1 is not considered prime. Any

natural number, no matter how large, can be factored into a product of primes.

Order Prime Order Prime Order Prime

1st 2 9th 23 17th 59

2nd 3 10th 29 18th 61

3rd 5 11th 31 19th 67

4th 7 12th 37 20th 71

5th 11 13th 41 21st 73

6th 13 14th 43 22nd 79

7th 17 15th 47 23rd 83

8th 19 16th 53 24th 89