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

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Intern
Joined: 06 Apr 2018
Posts: 5
How many different prime factors does 3^25 - 3^22 have?  [#permalink]

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14 Dec 2019, 00:19
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Difficulty:

25% (medium)

Question Stats:

68% (00:46) correct 32% (00:39) wrong based on 44 sessions

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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.

$$3^2^5 - 3^2^2$$

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

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

=$$3^2^2 (26)$$

=$$3^2^2 (2*13)$$
Math Expert
Joined: 02 Aug 2009
Posts: 8341
How many different prime factors does 3^25 - 3^22 have?  [#permalink]

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14 Dec 2019, 03:50
2
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
_________________
Intern
Joined: 06 Apr 2018
Posts: 5
How many different prime factors does 3^25 - 3^22 have?  [#permalink]

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14 Dec 2019, 18:05
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.
Math Expert
Joined: 02 Aug 2009
Posts: 8341
Re: How many different prime factors does 3^25 - 3^22 have?  [#permalink]

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14 Dec 2019, 20:10
1
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
_________________
Intern
Joined: 06 Apr 2018
Posts: 5
Re: How many different prime factors does 3^25 - 3^22 have?  [#permalink]

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14 Dec 2019, 23:59
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?
Re: How many different prime factors does 3^25 - 3^22 have?   [#permalink] 14 Dec 2019, 23:59
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# How many different prime factors does 3^25 - 3^22 have?

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