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Re: How many different prime factors does 3^25 - 3^22 have? [#permalink]
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tkbchimyjr18 wrote:
chetan2u wrote:
tkbchimyjr18 wrote:
How many different prime factors does \(3^{25} - 3^{22}\) have?

A) 1

B) 2

C) 3

D) 4

E) 5

I simplified the expression by factoring out at 3^22 and simplifying. But afterwards, I didn't know what to do.

\(3^2^5 - 3^2^2\)

= \(3^2^2 (3^3 - 1)\)

= \(3^2^2 (27 - 1)\)

=\(3^2^2 (26)\)

=\(3^2^2 (2*13)\)



\(3^{25}-3^{22}=3^2^2 (3^3 - 1)\)\(=3^2^2 (27 - 1)\)\(=3^2^2 (26)\)
\(=3^{22}(2*13)\)

So 3 different prime factors —— 2, 3 and 13

C



So I understand why 2, 3 & 13 are prime factors. However, I'm still confused. I don't understand why we're not looking to find the prime factors of the value equal to 3^2^2.


\(3^22\) is nothing but 3*3*3..., so it consists of only one prime factor 3.
For example 2^3 =8 and it has only one prime factor 2
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Re: How many different prime factors does 3^25 - 3^22 have? [#permalink]
chetan2u wrote:
tkbchimyjr18 wrote:
chetan2u wrote:
C



So I understand why 2, 3 & 13 are prime factors. However, I'm still confused. I don't understand why we're not looking to find the prime factors of the value equal to 3^2^2.


\(3^22\) is nothing but 3*3*3..., so it consists of only one prime factor 3.
For example 2^3 =8 and it has only one prime factor 2



Oh ok, so what's the rule here? It feels like there's some underlying rule/concept that I must not be aware of...Are you saying that any prime number raised to any exponent will only have itself as a prime factor? Is that the rule? A prime number raised to an exponent will never be divisible by another prime number?
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Re: How many different prime factors does 3^25 - 3^22 have? [#permalink]
\(3^{25} - 3^{22}\)
=\(3^{22}(3^3 - 1)\)
=\(3^{22} (26)\)
=\(3^{22} (2*13)\)

Different prime factors = 2,3,13

Hence, OA is (C).
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Re: How many different prime factors does 3^25 - 3^22 have? [#permalink]
Can someone explain why we don't try to break down 3^22 any further? I know 3 is a prime, but how can we be certain that it won't result in a different prime number without trying to open it up?
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Re: How many different prime factors does 3^25 - 3^22 have? [#permalink]
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Re: How many different prime factors does 3^25 - 3^22 have? [#permalink]
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