The GCF of Extremely Large Numbers

[Chill8160]

Hello!

I’ve been given a set of numbers:

5^2000 - 24*999 - 25
5^2002 - 24*1000 - 25
5^2004 - 24*1001 - 25

I have been challenged to find the Greatest Common Factor of these three values. Obviously, a calculator can not handle such large values.

I am unsure where to start!! Any help would be greatly appreciated!

Thanks in advance!


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[Deleted member 4993]

Hello!

I’ve been given a set of numbers:

5^2000 - 24*999 - 25
5^2002 - 24*1000 - 25
5^2004 - 24*1001 - 25

I have been challenged to find the Greatest Common Factor of these three values. Obviously, a calculator can not handle such large values.

I am unsure where to start!! Any help would be greatly appreciated!

Thanks in advance!


Sent from my iPhone using Tapatalk

What topic is being taught to you now?

 

[Dr.Peterson]

I’ve been given a set of numbers:

5^2000 - 24*999 - 25
5^2002 - 24*1000 - 25
5^2004 - 24*1001 - 25

I have been challenged to find the Greatest Common Factor of these three values. Obviously, a calculator can not handle such large values.

I am unsure where to start!! Any help would be greatly appreciated!

My first thought is that the GCF will divide the differences of the numbers. This is related to the Euclidean Algorithm.

 

[Chill8160]

What topic is being taught to you now?

I’m not currently taking a math course. I’m a senior in college and this was given to me by a computer science professor.

I’ve taken calc 1-3, discrete, linear algebra (matrices), stats 1-2, and all of the prerequisites for those courses.

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[stapel]

I’m not currently taking a math course. I’m a senior in college and this was given to me by a computer science professor.

I’ve taken calc 1-3, discrete, linear algebra (matrices), stats 1-2, and all of the prerequisites for those courses.

Can you write a program that can apply the Euclidean Algorithm to these numbers (and then check your answer here)?

 

[Dr.Peterson]

I’m not currently taking a math course. I’m a senior in college and this was given to me by a computer science professor.

I’ve taken calc 1-3, discrete, linear algebra (matrices), stats 1-2, and all of the prerequisites for those courses.

You may well have learned what you need in Discrete Math. Did you try using my hint about the differences of the numbers? What did you find those differences to be?

 

[Steven G]

Hello!

I’ve been given a set of numbers:

5^2000 - 24*999 - 25
5^2002 - 24*1000 - 25
5^2004 - 24*1001 - 25

I have been challenged to find the Greatest Common Factor of these three values. Obviously, a calculator can not handle such large values.

I am unsure where to start!! Any help would be greatly appreciated!

Thanks in advance!


Sent from my iPhone using Tapatalk

5^{large integers} will always end in 25. For example 5³ = 125.
So when you subtract the 25 you will have a multiple of 100. -24*999 = -24*1000 +24. So the first number ends in 24 and hence is divisible by 2 and 4.

The 2nd number is a multiple of 100.

5²⁰⁰⁴ -25 ends in double 0, ie 00. Then you subtract 24*1000 and the result still ends in 00. Now subtract that last 24 and the result will end in 76. So 2 and 4 goes into this number.

As a result, all 3 numbers are divisible by 2 and 4.

See if this helps.