Prime FactorizationDate: Thu, 26 Jan 1995 15:59:27 AST From: Richard Seguin Subject: Prime Factorization Could one of you guys give me a crash course on Prime Factorization. We just started that Section at school today, and since I come from a different province, I know nothing about it. In class they can divide so quickly, is there a trick to dividing 2 numbers so quickly? Date: 26 Jan 1995 15:59:31 -0500 From: Elizabeth Weber Subject: Re: Prime Factorization Hello there! Prime factorization is pretty simple once you know what's going on, and it's fun too! I used to do it in the margins of my notebooks when I was bored in class. Every whole number can be put into one of two categories: prime or composite. Prime numbers are numbers that you can't divide without getting a fraction, unless you feel like dividing them by themselves or by 1. These are numbers like 3, or 5, or 7, or 19, or 379721 (I told you I used to do this when I was bored!). Composite numbers, on the other hand, can be split up into something smaller; 8 is 2 times 4, 4 is 2 times 2, 21 is 3 times 7, 100 is 5 times 5 times 2 times 2, and so on. Thus, 8, 4, 21, and 100 are all composite numbers. It's easy to remember which word means which--composite has the word compose in it, and you can make composite numbers out of smaller numbers just like you can compose a piece of music out of smaller sounds. Now, the number 1 is a problem. I've heard people say that it's prime, since you can divide it by 1 without getting a fraction. I've also heard people say that it has its own little category, because a prime number should be divisible by two numbers: 1 AND itself, and nothing else; and since 1 is one, you can only divide it by 1 number without getting a fraction. I'd ask your teacher whether she thinks it's prime or not, but it doesn't really matter, at least not when you're doing prime factorization. All prime factorization is, is taking a composite number and splitting it up into the little numbers that it's made up of until you can't split it up any more. Take the number 8, for example. We can divide 8 into 2 and 4. 8 / \ 2 4 But we're not done yet; 4 can be split into 2 and 2 / \ 2 2 Now we're done. If you draw the smaller pieces (called the factors) in an upside down tree, like I just did, you can go back and collect all the pieces at the ends of the branches, and they will be the "prime factors" of the number. So, the prime factors of 8 are 2, 2, and 2. Here's another example: 30 30 / \ 5 6 / \ 2 3 So, the prime factors of 30 are 2, 3, and 5. Now, you wanted to know some tricks of dividing: 2--every even number is divisible by 2; if a number ends in a 2, 4, 6, 8, or 0, at least one of its factors will be a 2. 3--if you add the digits of a number, and the number you get is divisible by 3, then the original number is divisible by 3. For instance, if you take the number 57, and you add the digits, 5+7=12, and since 12 is divisible by 3, 57 is divisible by 3; 3 is a factor of 57. 4--take the last 2 digits of the number. If they are divisible by 4, then the number is divisible by 4. 216, for example, is divisible by 4, because 16 is divisible by 4. 5--anything that ends in a 5 or a 0 is divisible by 5. 6--anything that is divisible by 2 and by 3 is divisible by 6. Can you figure out why? 7--Somebody once told me that there was a shortcut to finding out if a number was divisible by 7, but she said that it would take just as long to walk to the store and buy a calculator as to use it, so it really wasn't much of a short-cut. 8--If the last 3 digits of the number are divisible by 8, the number is divisible by 8. 9--If the sum of the digits of the number is divisible by 9, then the number is divisible by nine. 10--If a number ends with a 0, it's divisible by 10. 12--If a number is divisible by 3 and by 4, it's divisible by 12. Can you figure out why? Those are all of the short-cuts to finding factors that I know. If you have any more questions, or if any of this doesn't make sense to you, write back to us! For summaries, see "Divisibility Rules" and "Explaining 3, 9, 11, 7, 13, 17, and larger numbers": http://mathforum.org/k12/mathtips/division.tips.html http://mathforum.org/k12/mathtips/ward2.html -Dr. Elizabeth The Math Forum Check out our Web site! http://mathforum.org/dr.math/ |
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