Greatest Common Factors and Least Common Multiples

I vaguely remember when I learned about factor trees and prime factorization for the first time, but they are definately important to finding the Greatest Common Factor (GCF) and the Least Common Multiple (LCM).  Though there is only one prime factorization of a number, there are 3 method for finding the prime factorization.
Remember a prime number has only 2 factors!
One method is increasing primes.  In this method you divide the number by increasing prime numbers.
Example:  84=2•2•3•7
               84÷2=42       42÷2=21      21÷3=7

There is the 2 factors method, where you keep writing numbers as a product of 2 factors.
Example:  84= 2•42
                       2•6•7
                       2•2•3•7
Then there is the factor tree, my favorite method.

Image from:  schoolworkout.co.uk

Above is an example of a factor tree for the number 72.  72 is broken down into 2 factors, 9 and 8, then 9 and 8 are broken down into 2 factors, and so on, until the numbers that are left are the prime numbers.  The numbers that are circled are the prime numbers. 

Once you understand prime factorization you can explore GCF and LCM.

GCF

GCF(a,b)= the largest number that is a factor of both a and b

Example:  GCF(7,14)=7

LCM

LCM(a,b)= the smallest number that is a multiple of both a and b

Example:  LCM(3,5)=15

Check out methods for finding the GCF and LCM at http://www.purplemath.com/modules/lcm_gcf.htm

Then you can test out your skills at http://www.math-play.com/Factors-and-Multiples-Jeopardy/Factors-and-Multiples-Jeopardy.html

The difference between multiples and factors

I was looking for a video on YouTube to make factors and multiples more fun, and this video made me giggle a little.  I like rhymes and songs that help me remember, even if they are a little juvenile.



          

Factors and Multiples

If a and b are whole numbers and a is not 0, then a is a factor of b, if and only if there is a whole number c such that a•c=b.  We can also say that a divides b or that b is a multiple of a.

Here is an example of  the 6 sentences that represent this:

• 2 is a factor of 20, since 2•10=20

• 10 is a factor of 20, since 10•2=20

• 2 divides 20, since 2•10=20

• 10 divides 20, since 10•2=20

• 20 is a multiple of 2, since 2•10=20

• 20 is a multiple of 10, since 10•2=20

A site that my help is http://www.kwiznet.com/p/takeQuiz.php?ChapterID=732

In class, we also covered divisibility tests.  I can't believe that I have never learned these tricks.

By 2:  a number is divisible by 2 if the ones digit is divisible by 2

By 5:  a number is divisible by 5 if the ones digit is a 5 or 0

By 3 or 9:  a number is divisible by 9 if the sum of its digits is divisible by 9, a number is divisible by 3 if the sum of its digits is divisible by 3

By 6:  a number is divisible by 6 if it is divisible by both 2 and 3

By 4:  a number is divisible by 4 if the last 2 digits are divisible by 4

By 8:  a number is divisible by 8 if the last 3 digits are divisible by 8   

I like the way that the divisibility tests are demonstrated on http://www.mathgoodies.com/lessons/vol3/divisibility.html

I also found the prime number test really helpful.  A number is a prime number if it has exactly 2 factors.  Once again, I can't believe that I have never learned this before.  The prime number test:

Suppose n is a whole number and k is the smallest whole number such that k•k>n (k²>n).  If there is no prime less than k that is a factor of n, then n is a prime number.

Example

51:  The largest prime you would need to check as a factor is the one that is ≤ √51≈7.1.  The largest prime ≤ 7.1 is 7.  So you would only need to check divisibility by 2,3,5,7.  It is divisible by 3, so 51 is not prime.

Until later.