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Is there any easy way to know what are prime factors of (b^n +-a)?

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Is there any easy way to know what are prime factors of (b^n +-a)?  [#permalink]

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New post 02 Oct 2018, 07:14
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For example imagine I have an integer y such that y = b^n +- c where b,n and c are integers. Is there any quick way to know what will the prime factors of y be without having to develop the number and then do a brute force prime factorization?

My understanding is that I can probably say is that c is going to "break" b^n... so if let's say b is 10 and we elevate to 2... we know that y will be 5*5*2*2 +/- c ... and then if c is 3 then I know that my new number will NOT be a multiple of 5 or 2 anymore... ok, but then what is it going to be a multiple of?

My understanding is that you could easily end up falling ON a prime after you add or substract c... so that leads me to think that there's no predictable way to figure out this case, hence one needs to do the brute force prime factorization and see what you get!

If anyone could shed some light here I'll be grateful! Thanks!
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Re: Is there any easy way to know what are prime factors of (b^n +-a)?  [#permalink]

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New post 07 Oct 2018, 11:55
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gmat800live wrote:
For example imagine I have an integer y such that y = b^n +- c where b,n and c are integers. Is there any quick way to know what will the prime factors of y be without having to develop the number and then do a brute force prime factorization?

My understanding is that I can probably say is that c is going to "break" b^n... so if let's say b is 10 and we elevate to 2... we know that y will be 5*5*2*2 +/- c ... and then if c is 3 then I know that my new number will NOT be a multiple of 5 or 2 anymore... ok, but then what is it going to be a multiple of?

My understanding is that you could easily end up falling ON a prime after you add or substract c... so that leads me to think that there's no predictable way to figure out this case, hence one needs to do the brute force prime factorization and see what you get!

If anyone could shed some light here I'll be grateful! Thanks!


This isn't a full answer to your question, but you can tell something about the prime factors relatively quickly. For instance, say that b = 10, and c = 6. You'd have something like

y = 10^50 + 6.

There's a rule about addition and factors, that says two things:
- If A and B share a factor, then A + B will also have that factor.
- If A has a factor, and B doesn't have that factor, then A + B doesn't have that factor.

In this case, you know that 10^50 has prime factors 2 and 5, while 6 has the prime factors 2 and 3. Therefore, you can conclude the following:

y has a prime factor of 2,
y DOESN'T have a prime factor of 3,
y DOESN'T have a prime factor of 5.

Unfortunately, I don't know a quick, consistent way to find every prime factor of y. But the information above is what I've seen used in GMAT problems in the past.
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Re: Is there any easy way to know what are prime factors of (b^n +-a)? &nbs [#permalink] 07 Oct 2018, 11:55
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