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# How many different prime factors does 3^25 - 3^22 have?

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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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