Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 4: Factors and Multiples 49

But how will you know when you’ve checked enough prime numbers? One way to tell is by com-

paring the quotient to the divisor. If the quotient is equal to or smaller than the divisor, you can

stop. You’ve checked enough. And if you get to the point where the quotient is less than or equal

to the divisor and you haven’t found a factor, your number is prime.

901 z 11 is 81 with 10 left over, and the quotient of 81 is larger than the divisor of 11, so this one

isn’t done yet.

901 z 13 = 69 with 4 left over, and the quotient of 69 is larger than the divisor of 13, so we keep

going.

901 z 17 = 53 exactly, which means 901 is a composite number. It can be written as 17 v 53 as

well as 901 v 1.

Consider Square Numbers

When you multiply any integer by itself, the result is a square number. The expression comes from

the fact that a square has the same length as width, so when you find its area, you multiply a

number by itself. The number 16 is a square number because it’s equal to 4^2 or 4 v 4, and that’s

the area of a square with a side 4 units long.

DEFINITION

Square numbers are numbers created by multiplying a number by itself, or raising

it to the second power. For example, 36 is a square number. It can be written as 6^2 or

6 v 6.

If you know the square numbers, it will help you see how far you have to go when checking

whether a number is prime. 30 v 30, or 302, is 900, which is very close to 901. The biggest prime

you’ll have to try is the one just above that, or 31. If you know your square numbers, it’s easy to

know how many primes you need to divide by before declaring a number prime. If you want to

know if 147 is prime, think about square numbers near 147. The number 144 is 12^2 , so if you go

to the next prime after 12, that’s enough. Once you divide 147 by 13, if you haven’t found a factor,

you can say 147 is prime.

Suppose you want to know if 67 is prime. If you know that 8^2 is 64, you only have to check 2, 3,

5, 7, and maybe 11. The number 67 is not even, so it’s not divisible by 2. Its digits add to 13, so it’s

not divisible by 3. It doesn’t end in 5 or 0, so it’s not a multiple of 5. When you divide 67 by 7, you

get 9 with 4 left over, so 7 is not a factor. 67 z 11 is 6 (smaller than the divisor) but with 1 left

over. 67 is a prime.