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### 3 Methods for Finding Least Common Denominator

```
Date: 02/04/98 at 10:25:37
From: Vera Rooney
Subject: Finding least common denominator

My fifth graders are having a very difficult time understanding how to
get the least common denominator.  Any shortcuts?  Thank you.
```

```
Date: 02/04/98 at 16:03:11
From: Doctor Rob
Subject: Re: Finding least common denominator

Shortcuts, no.  I use three methods when faced with a problem like
this.

First of all, the least common denominator is a multiple of all the
denominators. In fact it is the least common multiple (LCM) of all the
denominators. One can find the LCM of a set of numbers by doing it two
numbers at a time. Find the LCM of the first two numbers. Then find
the LCM of that result and the third number. That will be the LCM of
the first three numbers. Now find the LCM of this result and the
fourth number, and so on. This reduces the problem to dealing with
just two numbers at a time.

First method:
Write down the multiples of the two numbers in two lists. Find the
numbers which are on both lists, and pick the smallest.

Example: LCM of 12 and 18. Multiples of 12 are {12, 24, 36, 48, 60,
72, ...}.  Multiples of 18 are {18, 36, 54, 72, ...}.  Numbers common
to both lists are {36, 72, ...}.  The smallest is 36, the answer.

Second method:
Factor the two numbers into products of powers of prime numbers.
Create a new number multiplying together all the primes occurring in
either number, raised to the higher of the two exponents.  That is the
LCM.

Example:  LCM of 45 and 12.  45 = 3^2*5, 12 = 2^2*3.
New number = 2^2*3^2*5 = 180, the LCM sought.

Third method:
Find the greatest common divisor of the two numbers. Multiply the two
numbers together and divide by the greatest common divisor. (You have
to know how to find the greatest common divisor to be able to use this
method. Do your students know how to do that?)

Example:  LCM of 45 and 12.  Greatest common divisor is 3.  LCM is
45*12/3 = 180.

I hope this helps.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
Middle School Fractions

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