Greatest common divisor

matqkks

Are there any real life applications of the greatest common divisor of two or more integers?

Simon Bridge

Homework Helper
Yes. i.e. whenever you have something depending on a ratio ...

It's not normally expressed in that way though.
Mostly - the lesson is important for the practise it gives in a kind of problem solving.

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rexregisanimi

There are a whole bunch. I use the idea regularly so it is difficult to point to a specific thing...

Simon Bridge

Homework Helper
There are a whole bunch. I use the idea regularly so it is difficult to point to a specific thing...
Aww go on - show us one... what's the one you use regularly that you last used?

Permanence

Here take a look at this Wikipedia article.
https://en.wikipedia.org/wiki/Euclidean_algorithm

Besides all the mathematical applications, I'm sure that you must sometimes call upon the GCD when shopping or possibly even cooking.

• 1 person

rcgldr

Homework Helper
Are there any real life applications of the greatest common divisor of two or more integers?
I can't think of any real world applications. The closest I can think of is the Euclid algorithm for finding the GCD which can be extended and used to find the inverse of a number in finite field, but it's seldom used because there are other and better methods. For example, if the field isn't very large, a lookup table can be used. In the case of hardware implementations of inversion based on "binary" finite fields (which is part of AES encryption), there are complex methods (sub-field mapping) that involve fewer gates than a lookup table. Wiki article for extended Euclid algorithm:

wiki_inverse_in_finite_field.htm

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• 1 person

dm164

GCD is used any time you want to simplify integers, but have the same ratios as the others have said. Another way to say it is the numbers scale equally. An example is in solving empirical formulas, where you reduce all integers in a chemical formula. Such as hexane C6H8 -> C3H4 GCD(6,8) is 2 so divide each by 2 to get the answer. Though it isn't of much use, but it has a name.

• 1 person

economicsnerd

The store sells 8-packs of hotdogs and 12-packs of buns. If you want the same (nonzero) number of each, what's the cheapest way to do it?

Simon Bridge

Homework Helper
@economicsnerd: good example, well done!
Most people wouldn't do that by listing the divisors, but I suppose there are examples less amenable to a bit of trial and error.
Do HS math text books no longer have examples like that these days?

rcgldr

Homework Helper
The store sells 8-packs of hotdogs and 12-packs of buns. If you want the same (nonzero) number of each, what's the cheapest way to do it?
Wouldn't this be the "lowest common multiple" as opposed to the "greatest common divisor"?

economicsnerd

Wouldn't this be the "lowest common multiple" as opposed to the "greatest common divisor"?
Directly, I guess it's not really either.
Computing either the GCD or the LCM would get you close to knowing how many bags of each to buy. You either compute {h/GCD(h,b), b/GCD(h,b)} or {LCM(h,b)/h, LCM(h,b)/b}.

Stephen Tashi

If you can think of a real life application of linear Diophantine equations, then the GCD and the Euclidean algorithm have applications to solving those.

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