Reducing Fractions

Date: 03/17/97 at 21:19:24
From: SARAH WILLIAMS
Subject: Make Sure Fraction is Reduced

Dear Dr. Math,

How do you check/verify that a fraction is in its simplest form? 
For example, how do you know that 5/7 is reduced?

Thanks,
	Sarah

Date: 03/18/97 at 08:23:23
From: Doctor Jerry
Subject: Re: Make Sure Fraction is Reduced

Hi Sarah, 

The answer is that you use what is often called The Fundamental 
Theorem of Arithmetic, which is: 

  Every integer can be expressed as the product of one or more 
  prime numbers (2,3,5,7,11,13,... are prime numbers;  
  4,6,8,9,10,12,... aren't) to a power.  

An integer is a prime if it is divisible (without remainder) by only 
itself and 1.  So, 32, for example, can be written as 2^5 (this means 
2*2*2*2*2); 100 is 2^2 * 5^2 and so on.

To see if a fraction m/n is in its simplest form, apply the 
Fundamental Theorem of Arithmetic (FTA) to each of m and n (m is the 
top part of the fraction, n the bottom).  If there are no primes in 
common to m and n, then m/n is in simplest form. 

Here are two examples:

49/1547 = 7^2/(7*13*17), not in simplest form.  But FTA shows hows to 
put this fraction in simplest form.  Just cancel the extra 7.

49/1547= 7/(13*17) = 7/221.

7/247 = 7/(13*19), no common factors and so in simplest form.

Of course, for big numbers it's not so easy to factor a number into 
its prime factors.

You can now see that 5/7 is in its simplest form.  Both 5 and 7 are 
primes.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/