Factoring and Divisibility Rules
Date: 12/18/97 at 19:41:10 From: Kate Subject: Factoring I have no idea how to factor these problems: 12*13-60+12(squared) 11(squared)-6*11+5*11 I'm not sure where to start.
Date: 01/10/98 at 10:52:02 From: Doctor Marko Subject: Re: Factoring Kate, The word factoring is used in two contexts: 1. Factoring a number into its prime factors (like for example 12 = 2*2*3) 2. Factoring the expression such as the ones you have above so that you simplify the expression so you can do calculations with it more easily. In order to learn this one needs to know how to factor a number into its prime factors (see no. 1) and so I will explain that as well, even if it is only a review. If your question concerns factoring in the sense of no. 1, the recipe is to know the 'rules' of divisibility: i) even numbers are divisible by 2. ii) if the sum of the number's digits is divisible by 3 then that number is divisible by 3. iii) if the last two digits of a number are divisible by 4 then the number is divisible by 4. iv) if the number ends in 5 or 0 it is divisible by 5. Let's try number 105. You may be able to automatically say that it is divisible by 5, but it is customary to start from 2 and work your way up. So 2? No, the number is odd. 3? Well, 1+0+5 = 6, which is divisible by 3, and so 105 is divisible by 3. 105/3 = 35. (REMEMBER 3!) Now we have factored 105 into 3 and 35, but 35 is not a prime number, so we start the whole factoring procedure again, only now for 35. Is 35 divisible by 2? No, because it is odd. How about by 3? No, it is not, since the sum of its digits (3+5 = 8) is not divisible by 3. It is not divisible by 4 either, since it is not divisible by 2, but it is divisible by 5, since 35 ends in a 5. 35/5 = 7. (REMEMBER 5!) 7 is a prime number also, so we are done - we have the prime factorization of 105, which is 3*5*7. Good. It becomes harder when the prime factorization has larger prime numbers like 19 and 37, but in that case I am not sure that there is a safe recipe other than luck. However, it sounds like you are more interested in the no. 2 connotation of the word factorization. For that, follow these steps: a) Determine the biggest common factor among your numbers. Here the emphasis need not be on THE BIGGEST, but the bigger it is the easier it will be for you to factor. -- So it is clear that in your second example it must be 11, since every number in the sum is written as 11 times something. In the first case it is a bit tougher. The first thing you need to do is figure out the prime factorization of 60 by the method in no. 1. The factorization is 2*2*3*5, which you can write as 12*5, as well. So, now you have the same situation as in the second example, and clearly your biggest common factor is 12. b) Figure out the remaining factors of each of your numbers: -- In the first case those are 13, 5, and 12. And in case two they are 11, 6, and 5. c) Now you are ready to factor: -- First write your common factor which will be multiplied by a bracket. Inside the bracket you will have the remaining factors in the correct order and with the right signs. Then add the remaining factors and you are done. To make it more concrete let me completely work out one of your examples: 12*13 - 60 + 12*12 Common factor is 12, so 12*(13 - 5 + 12) = 12*20 = 240 Here you can see the power of factorization, since we solved that messy problem without doing much multiplication. Hope this has helped. If you need some additional help, please do not hesitate to write. -Doctor Marko, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.