Least Common Denominator Using Primes:

A prime number is a whole number (integer) whose only factors are itself and one. So the first prime numbers are given as follows:

1,   2,   3,   5,   7,   11,   13,   17,   19,   23,   29, . . . .

By dividing by primes, you can search that the primes of 105 are:

105/3 = 35    35/5 = 7 = a prime number, therefore, stop dividing.

The primes of 105 are:  3, 5, 7

A systematic way of searching the prime factors of larger positive integers is described below.  The primes are tried in order, as factors, using each as several times as possible before going on to the further.  The result in this case is:

504      =(2)(252)

=(2)(2)(126)

=(2)(2)(2)(63)

=(2)(2)(2)(3)(21)

=(2)(2)(2)(3)(3)(7)

To add various fractions with different denominators, subsequent these steps:

Step 1:  State denominators in prime factors.

Step 2:  Determine  the  least  general  denominator  by  using  all  of  the  prime numbers  from  the  largest  denominator,  and  then  involve  each  prime number from the other denominators so in which each denominator could be calculated from the list of primes contained in the LCD.

Step 3: Rewrite using the least general denominator.

Step 4: Add the fractions.