Algebra Know-It-ALL

the quotient are both factors of the original number. Sometimes the quotient will be prime,

and sometimes it won’t be. If it isn’t prime, then it can be factored down further. Don’t stop

dividing the original by primes until you get all the way down to 2.

Once you’ve found all the primes smaller than s that divide your original number without

giving you a remainder, you have found the prime factors of your original number. Some

product of these will give you that original number. You might find that one of the primes is

a factor twice, three times, or more. For example:

99 = 11 × 3 × 3

297 = 11 × 3 × 3 × 3

891 = 11 × 3 × 3 × 3 × 3

2,673= 11 × 3 × 3 × 3 × 3 × 3

Perfect squares

Whenever you multiply a number by itself, the process is called squaring, and the product is

called a square. If you multiply a natural number by itself, you get a perfect square. If you take

the square root of a perfect square, you always get a natural number.

Perfect squares, unlike primes, are easy to find if you have a calculator. Table 3-2 lists the

first 24 perfect squares, with 0 and 1 included.

Are you confused?

If you see a large natural number, you can sometimes tell right away that it’s not prime. If its numeral

ends in an even number, you know that one of its factors is 2, so it can be factored into a pair of natural

numbers other than 1 and itself.

Sometimes, numbers seem at first as if they ought to be prime, but it turns out that they are not. A

good example is 39. It can be factored into 13 × 3. Another is 51, which can be factored into 17 × 3. Still

another is 57, which can be factored into 19 × 3.

Table 3-2. The first 24 perfect squares. The numbers 0 and 1 are included here.

When you take the square root of a perfect square, you always get a natural number.

Note that we start with the “0th” rather than the “1st” in order here. That way, the

order agrees with the number squared.

Order Square Order Square Order Square

0th 0 8th 64 16th 256
1st 1 9th 81 17th 289
2nd 4 10th 100 18th 324
3rd 9 11th 121 19th 361
4th 16 12th 144 20th 400
5th 25 13th 169 21st 441
6th 36 14th 196 22nd 484
7th 49 15th 225 23rd 529

Special Natural Numbers 41