Re: reducing large fractions
From: Thomas Mautsch (mautsch_at_math.ethz.ch)
Date: 07/14/04
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Date: 14 Jul 2004 18:02:02 +0100
In news:<20040714012859.06753.00001460@mb-m13.aol.com> schrieb Mensanator:
>>Subject: reducing large fractions
>>From: "slag" rob@robfindlay.org
>>Date: 7/13/04 11:59 PM Central Daylight Time
>>Message-id: <cd2eip$5ks@odbk17.prod.google.com>
>>
>>Is there any shortcuts for reducing LARGE fractions.
>>
>>For instance: 6809/10800 ?
>
> Factor the numerator and denominator and cancel common factors.
^^^^^^^^
For large denominator and numerator, factoring will be overkill:
> Nevertheless, it can still be factored: 6809 = 11 * 619
>
>>so it wont reduce?
>
> You'll need to check the factors of 10800: 2^4 * 3^3 * 5^2
> Note that the numerator and denominator have no common
> factors, so you cannot reduce the fraction.
The better alternative would be to use Euclid's algorithm
for the greatest commen divisor of 10800 and 6809:
10800 : 6809 = 1 R. 3991
6801 : 3991 = 1 R. 2818
3991 : 2818 = 1 R. 1173
2818 : 1173 = 2 R. 472
1173 : 472 = 2 R. 229
472 : 229 = 2 R. 14
229 : 14 =16 R. 5
14 : 5 = 2 R. 4
5 : 4 = 1 R. 1
4 : 1 = 4 ===
So the g.c.d. is 1, and the fraction can not be reduced.
This was not as effortless as expected,
so what one might rather do is
to split off as many prime factors from numerator and denominator
as one is able to find,
and then find the g.c.d. of the remaining terms.
(As an application of the rule: gcd(a,b*c) = gcd(a,c) when gcd(a,b)=1.)
For the example 6809/10800,
finding the complete factorisation of 10800 is very easy,
and 6809 is not divisible by either of the prime factors 2, 3, and 5
of 10800,
but if, for example, one only knew that
10800 is 100 * 108 = 2^2 * 5^2 * 108,
and that the prime factors 2 and 5 are no divisor of 6809,
then one would calculate the g.c.d. of 6809 and 108:
6809 : 108 = 63 R. 5
108 : 5 = 21 R. 3,
hence:
gcd(10800,6809) = gcd(6809,108) = gcd(5,3) = 1.
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