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i need help with my homework with gcf with distributive property

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3 Answers

 Right the numbers as product of their prime numbers.
 
   18 = 2 .  32
  
     12  = 22 . 3
   
     GCF = product of common factors with lower exponents = 2 * 3 = 6  ( greatest number that both numers
                                                                                                            can be divided by)
 
       LCM = product of common factors with highest exponents = 22 . 32 = 4 * 9 = 36 (Greatest number that
                                                                                                                            both  numbers can     
                                                                                                                         be  divided into that )                                                                                            
Consider the relationship between greatest common factor and least common multiple.
 
Let's look at 18 and 12. 
 
The prime factorization of 18 is 2x3x3, and the prime factorization of 12 is 2x2x3. Then, a multiple of 18 that is also a multiple of 12 is 2x2x3x3 [(2x3x3x2 = 18x2) and 2x2x3x3 (2x2x2x3 = 12x3)], or 36.
 
But, this is just the product of 12 and 18 (216) divided by least common multiple of 12 and 18 (6). 216/6 = 36.
 
A general method is LCM(a,b) = (axb)/GCF(a,b), and GCF(a,b) = (axb)/LCM(a,b).
 
This method will be much easier when dealing with polynomials in high school. 
Express the sum of each pair of numbers as a product of the greatest common
factor of the numbers and another sum.
 
so factoring 18 = 2x3x3 = 3x6
factoring 12= 2x2x3 = 2x6
 
The greatest common factor between the two is 2x3 = 6
GC(18,12) = 6
 
18+12 = 6x3+6x2 = 6(3+2)