1.9+Prime+Numbers

Section 1.9 - Prime Numbers

 * Prime vs. Composite**

Let's start with the definitions:

Prime Number - Has no factors except 1 and itself. Examples are 2, 3, 5, 7, 11. None of these can evenly be divided by any numbers except 1 and itself.

Composite Number - A number that has more factors than just 1 and itself. Examples are 4, 8, 15, 90. Factors of 15 are 1, 3, 5, and 15.

Because 2 goes evenly into all even numbers, 2 is the only prime number that is also even. All other prime numbers are odd, but not all odd numbers are prime!


 * 0 and 1 are Neither**

Even though they seem to follow the rule for prime numbers, 0 and 1 are considered neither prime, nor composite. They don't get their own category title, they just are neither.


 * Divisibility Rules**

The divisibility rules can help to find factors of a number.

2 - If a number is even, 2 will be a factor. The number is even if it ends in 0, 2, 4, 6, or 8.

3 - If you add up the digits of a number and the sum is divisible by 3, the whole number is divisible by 3. For example, if you add up the digits in 2,316, you get 12 (2+3+1+6 = 12), which is divisible by 3 (3 x 4 = 12). That means that 2,316 is also divisible by 3. Three time something = 2,316. It might require long division to figure out that 3 x 772 = 2,316.

4 - If the last two digits in the number are divisible by 4, the whole number is. You don't add those last two digits up, you treat them like a two digit number. 54,621 is not divisible by 4, because 21 (last two digits) is not.

5 - All numbers ending in either 5 or 0 are divisible by 5.

10 - If it ends in a 0, it is divisible by 10.

There are other rules for other digits as well, but I don't think they are overly important. 7 is very complicated, and 6 and 9 are multiples of other, easier digits anyway.


 * Prime Factorization**

Prime factorization means reducing a number to a product of only prime factors. The answer will be written in ascending order of factors. For example, 40 could be written as 2 x 2 x 2 x 5, because each of those is a prime number, and when they are multiplied, they equal 40. Another (better) way to write them is with exponents:



There are a couple of ways to do prime factorization. One common way is known as the factor tree.




 * Using Repeated Division to do Prime Factorization**

Another method, not as fancy maybe, but easy, is to just repeatedly divide the number by prime factors, keeping track of those factors. In practice, this looks a lot like that second tree example.

60 / 2 = 30 / 2 = 15 / 3 = 5 So the factors are 2, 2, 3, and 5. Multiply them all together and you get 60.

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**Where to from here?**
1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Chapter One Review**
 * Chapter One Homework**