# Greatest common factor with exponents of first input

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1. Jan 13, 2016

### DirichletHole

ok so is there a function that exists (for all intents and purposes let's call it G(x,y) )where

x= a^2*b^4*c
y=a^4*b^2*d

G(x,y) = a^2*b^4

basically gcd, but the exponents match those of the common prime factors of the first input (x)

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equally useful would be a function where the output would be the coprime factor(s) of the first input. ie
B(x,y) = c

2. Jan 14, 2016

### Staff: Mentor

Actually the result would be $a^2b^2$.

A better way to write this would be $G(a^2b^4c, a^4b^2d) = a^2b^2$.
Is there such a function? I assume your question is, "Is there a function in some programming language that returns the GCF of its two inputs?"
I don't know, but possibly someone has written such a library function in, say, the numpy or scipy Python libraries, or some other library in some other programming language.

3. Jan 14, 2016

### DirichletHole

ok let me be more specific:

first off i am looking for a mathematical algorithm, something that could exist outside of a computer program

secondly, the equation given was purely just an example. What I need is an algorithm "G(x,y)" where, given two functions (a first function "x" and a second function "y"), G(x,y) returns a number who has all and only prime factors that are the shared prime factors of x and y, but are taken to the greatest exponents that still divide x (the first of the two functions plugged into G)

alternatively, equally useful to me would be an algorithm "B(x,y)" where, given the same two functions x and y, B(x,y) returns a number that is divisible by all and only the prime factors of x (with the exponents that they have as divisors of x) that are not shared with y.

4. Jan 14, 2016

### Staff: Mentor

I'm not aware of anything like this. Are you saying that, for example, $G(2^{10}, 2) = 2^{10}$?
I don't know how useful such an algorithm would be, but all an algorithm is, is a sequence of steps to be performed. Assuming my example follows what you're describing, how would you explain to someone how a result is achieved? That would be your algorithm.

Last edited: Jan 14, 2016
5. Jan 14, 2016

### DirichletHole

Yeah exactly: I'm looking for a function G(x,y) that would work exactly like the example you gave. G(2^10, 2) = 2^10.

The Algorithm would be extremely useful to me personally. I'm looking for as nice and clean and simple a function as possible. The best example i can think of so far would be G(x,y) = gcd(x,y^x), but i'd prefer something more precise, and something that would not require processing unnecessarily large numbers, as y^x would get very very large.

If such a G(x,y) can not be easily and cleanly created, a function that would be of equal use to me would be a function B(x,y) that would return the divisors of x that are coprime to y. For example B(2^10*3^5*5^8*7^3*11, 2^3*3^2*7^2) = 5^8*11

If it is at all helpful, x will always be a factorial and y will always be made up of factors less than or equal to x. So x will have prime factors that y won't have, but y will only be made up of prime factors that x also has. Also the exponents on the prime factors of y will *probably* always be less then the exponents of the prime factors of x.

Last edited by a moderator: Jan 14, 2016