IN MATH: 1. n. a denomimator (divisor, bottom number) shared by two or more fractions. EX. Common denominators are needed to add fractions.
A common demonimator is not just the product of the factors the fractions have in common but, THE FACTORS THEY HAVE IN COMMOM AND THE FACTORS THAT MAKE EACH FRACTION DIFFERENT. A common demonimator includes all factors from each fraction and there are many common denominators.
The LEAST COMMON DENOMINATOR is the smallest product of the factors of each fractions .
Here are two methods for FINDING THE LEAST COMMON DENOMINATOR, the smallest product of the factors of each fractions .
In the examples below, the lines starting at the top of the example show the first step in the work.
Ex. 1. Find the common denominator for 5/12 and 3/4. Common Factors First The common factors are 2 and 2. The first denominator also has a 3. The common denominator is 2·2·3 or 12.  One Fraction AtATime The first denominator has factors of 2 and 2 and 3. The second denominator has factors of 2 and 2 and these are already listed. No new factors need be listed. 
Common Factors First There are no common factors. List each fraction's factors. The common denominator is 2·2·3 or 12.  One Fraction AtATime The only prime factor of 3 is 3. Next, list the factors of 4. The common denominator is 2·2·3 or 12. 
Common Factors First There is no common factor of 6, 2, 12, and 15. Next most common is 2 and 3, so list these avoiding duplication. An additional 2 and an additional 5 are required. The common denominator is 2·2·3·5 or 60.  One Fraction AtATime The prime factors of 6 are 2 and 3, so, list each. The prime factor of 2 is 2 but, it has already been listed. The prime factors of 12 are 2 and 2 and 3. The 2 and 3 are duplicates, so, just list one more 2. The prime factors of 15 are 3 and 5. List the 5, but, don't duplicate the 3 which is already listed. The common denominator is 2·2·3·5 or 60. 
Common Factors First Ten and 25 share a common factor of 5, so, list it. The additional factors of 2 and 5 are required. The common denominator is 2·5·5 or 50.  One Fraction AtATime List the factors 2 and 5 for the denominator 10. Don't list the duplicate factor of 5, but, do list an additional factor of 5 for the factors of 25. The common denominator is 2·5·5 or 50. 
Example 5 is very much like example 4. One has arithmetic factors, the other algebraic factors. The procudure is the same.
Ex. 5. Find the common denominator for 2/x^{2} and 3/(xy). Common Factors First Each denominator has a factor of x. List it only once. Next list the two required factors of x and y. The common denominator is x·x·y or x^{2}y.  One Fraction AtATime The first fraction has denominator factors of x and x. List each. The next fraction has factors of x and y. Do not duplicate the x. List the y. The common denominator is x·x·y or x^{2}y. 
Common Factors First The binomials 4x+4y and 2x2y have only 2 as a common factor. List it. Other unique factors needed are x+y, xy, and 2. List each. The common denominator is 2·2·(x+y)·(xy) or 4x^{2}4y^{2}.  One Fraction AtATime The first fraction has a denominator prime factors 2, 2, and x+y. List these. The second denominator has a duplicate factor of 2 and a unique factor of xy. List only the xy. The common denominator is is 2·2·(x+y)·(xy) or 4x^{2}4y^{2}. 
Common Factors First The fractions have a common denominator factor of x+1. List it. They have unique factors of x1 and x+3. List them. The common denominator is (x+1)(x1)(x+3).  One Fraction AtATime The first fraction as denominator factors of x+1 and x1. List them. In the second fraction's denominator the x+1 is a duplicate. Don't list it. The x+3 is needed. List it. The common denominator is (x+1)(x1)(x+3). 
Common Factors First The denominators have a common factor of x+4. List it. Unique factors of 2 and x4 are also needed. List them. The common denominator is 2(x+4)(x4).  One Fraction AtATime The first fraction has denominator factors of 2 and x+4. List them. The second fraction has a duplicate factor of x+4. Don't list it. It has a unique factor of x4. List it. The common denominator is 2(x+4)(x4). 
Common Factors First The common factors are 3, a, b, and b. List them. There are unique factors of 2, 2, a, 3, and b. List these also. The common denominator is 2·2·3·3·a·a·b·b·b or 36a^{2}b^{3}.  One Fraction AtATime The first denominator has factors of 3, 3, a, b, and b. List them. The next fraction has duplicate factors of 3, a, b, and b. Do not list these. List the unique factors of 2, 2, and b. The common denominator is 2·2·3·3·a·a·b·b·b or 36a^{2}b^{3}. 

IN ENGLISH: 1. n. a characteristic shared by two or more things or people. EX. Their common denominator was their love for mathematics.
APPLICATION: See list 40.
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