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How do I find the (LCM) or the (LCD) of fraction?

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2 Answers

Yet another way, which I mentioned to a number of students, is by first finding the GCD (greatest common divisor) and using the fact that LCM(x,y) * GCD(x,y) = xy.

The GCD can be found by Euclid's algorithm which proceeds by successively subtracting multiples of the smaller argument from the larger one. The base case is GCD(n,0) = GCD(0,n) = n. For large arguments, this is the fastest method possible.

Example:

x = 632, y = 412

GCD(632,412) = GCD(220,412) since 632 - 412 = 220.

                     = GCD(220,192) since 412 - 220 = 192.

                     = GCD(28,192)   since 220 - 192 = 28.

                     = GCD(28,24)     since 192 - 6*28 = 24.

                     = GCD(4,24)      since  28 - 24 = 4.

                     = GCD(4,0)        since 24 - 6*4 = 0

                     = 4 using the base case.

Now for the LCM:

LCM(632,412) = 632*412 / GCD(632,412) = 632*412/4 = 65096.

The nice thing about this method is that you don't need prime factorizations of the two numbers, which are very hard to find if the numbers are very large with very large prime factors.

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Exercise for you:

Use the interpretations of GCD(x,y) and LCM(x,y) in terms of prime factorizations of x and y to prove that LCM(x,y) * GCD(x,y) = xy.

Hint: You will want to show that Min(a,b) + Max(a,b) = a + b.

Yes they are the same number, but LCM, for Least Common Multiple is a more descriptive term, as it hints at a way to find it.

To find these, one could list the multiples of each number (or denominator).

For example, say one needs the LCM of 6 and 15.

6:  6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, etc.

15:  15, 30, 45, 60, 75, etc.

30 and 60 are the two smallest common multiples, but the least of all is 30.

So 30 is the LCM of 6 and 15.

If two fractions had 6 and 15 as their respective denominators, one would say that their LCD, for Least Common Denominator, would be 30.  This means both fractions could be rewritten as fractions with 30 as the denominator. 

A / 6 + B / 15 = 5A / 30 + 3B / 30.

There is another way to find the LCM [ or LCD] involving prime factorizations, but I will leave that for you to discover on your own.  It can actually be easier when the numbers involved are large.

Hope this helps.  If you have any more questions, please feel free to ask.  I or another tutor will be happy to assist you.