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# For every positive integer x, f(x) represents the greatest prime fact

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Manager
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 11:28
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For every positive integer xx, f(x)f(x) represents the greatest prime factor of x!x!, and g(x)g(x) represents the smallest prime factor of 2x+12x+1. What is (g(f(12))(g(f(12))?

A. 2
B. 3
C. 5
D. 7
E. 11

f (12 ) is greatest prime of 12!, which is 11.

so g (11) =smallest prime of 2^11 +1 so 2048 + 1 ( infact 2 power odd +1 then 3 will be a divisor)= 2049 is divisible by 3 and not by 2. so 3 is smallest prime

so ans is B
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 11:52
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For every positive integer $$x$$, $$f(x)$$ represents the greatest prime factor of $$x!$$, and $$g(x)$$ represents the smallest prime factor of $$2^x+1$$. What is $$(g(f(12))$$?

f(x) = greatest prime factor of (x!)
f(12) = greatest prime factor of (12!) --> greatest prime factor of (12x11x10x...x2x1) = 11

g(x) = smallest prime factor of (2^x+1)
g(f(12)) = g(11) = smallest prime factor of (2^11+1) = smallest prime factor of (2049) = smallest prime factor of (3*683) = 3

Therefore g(f(12)) = 3.

A. 2
B. 3
C. 5
D. 7
E. 11

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 12:55
1
Very nice and typical question for GMAT.

First of all, we need to figure out the greatest prime factor of 12!
It's clear, that 11 is the greatest, because 12 = 3*2*2.

The next step is a little bit harder.
2^11 +1
It's clear that this number is not divisible by 2, so A is out.
Unfortunately, this information is not enough.

2^11 + 1 = 2^5 * 2^5 * 2 + 1= 32*32*2 + 1= 2049
The sum of the digits of this number (2+4+9 = 15) is divisible by 3.
It means that the number itself is divisible by 3.

So the smallest prime factor of 2049 is 3

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 13:36
1
So f(x) = greatest prime factor of x!
g(x)= smallest prime factor of (2^x+1)

f(12) = 1•2•3•4•5•6•7•8•9•10•11•12
Greatest prime factor =11
.: f(12)=11

g(11) = 2^11 + 1 => 2^5•2^6+1
g(11) = 2049 = 3•683
Smallest prime factor =3
.: g(f(12) =3

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 13:48
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f(12) : highest prime factor of 12! is 11, so f(12)= 11
G (11) = 2^11+1, which can be written as 32*32*2 +1 : So lowest factor would be 2+1 =3
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 14:33
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The largest prime factor for $$12!$$ is $$11$$, so$$f(12) = 11$$
if $$x = 11$$, then$$2^{11} + 1 = 2048 + 1 = 2049$$, not divisible by 2, but divisible by 3, so $$(g(f(12)) = 3$$

B
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 15:27
1
f(12) = greatest prime of 12! is 11.
g(11) = smallest prime of (2^11 + 1)

Since 2^10 = 1024, 2^11 = 2048.
Therefore g(11) = 2049.

Since its odd number, check the next smallest prime factor answer choice
Now, 2+4+9 = 15, 2049 is divisible by 3.

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 17:13
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f(x) represents the greatest prime factor of x!.
12!=1*2*3*4*5......9*10*11*12
Distinct Prime factors of 12! are 2,3,5,7 and 11. Hence the greatest prime factor of 12! is 11

f(12)=11

g(x) represents the smallest prime factor of $$2^x+1$$
g(11) is the smallest prime factor of $$2^{11}+1$$

$$2^{11}$$+1 = 1 (Mod 2). Hence $$2^x+1$$ is not divisible by 2.

$$2^{11}$$= $$(-1)^{11}$$ Mod 3
$$2^{11}+1$$= [(-1)+1] Mod 3
$$2^{11}$$+1= 0 Mod 3
$$2^{11}+1$$ is divisible by 3. Hence the smallest prime factor of $$2^{11}+1$$ is 3.

(g(f(12))=3

IMO B
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For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 18:16
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$$f(12) =$$ the greatest prime factor of $$12! = 12*11*10*9*...*1$$ --> $$f(12) = 11$$.

$$g(f(12)) = g(11) =$$ the smallest prime factor of $$(2^{11}+1) =$$ the smallest prime factor of $$2049.$$

Since $$2049 = 3 * 683$$, where 3 and 683 are both prime factors, then $$g(f(12)) = g(11) = 3.$$

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For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 20:26
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f(12) = 12! ==> greatest prime factor of 12! = 11

g(11) = $$2^{11}$$+1 = 2048 + 1 = 2049 ==> 2+0+4+9 = 15, which is divisible by 3 ==> smallest prime factor of $$2^{11}$$ + 1 = 3 (choice B)
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 20:32
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(g(f(12)))
f(12)= greatest prime factor of 12! This means the highest prime number from 1 to 12 which is 7.
Hence f(12)=7
Now g(7) is the least prime factor of ((2^7)+1)
(2^7)+1 = 128+1= 129.
Prime factors of 129 are
{3 and13}.
Hence g(7)=3.

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 21:14
1
For every positive integer x, f(x)represents the greatest prime factor of x!, and g(x) represents the smallest prime factor of 2^x+1. What is (g(f(12))?

A. 2
B. 3
C. 5
D. 7
E. 11

Given (g(f(12))
f(12) = highest prime factor of 12! will be 11 since there will no prime factors greater than 11
now g(11) = smallest prime factor of$$2^{11}$$ +1

$$2^{11}$$ will be even and +1 will make it odd
now $$2^{11}$$ +1 = 2048 +1 =2049
this is completely divisble by 3 giving 683
thus smallest is 3 hence B
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 21:22
1
Given: x is an integer, f(x)=greatest prime factor of x!, g(x)=smallest prime factor of (2^x)+1
To find: g(f(12))=?

Solution: For f(12), x=12 so prime factors for 12! are 2,3,5,7,11
therefore, greatest prime factor=11. Hence f(12)=11
now,
g(f(12))=g(11)
g(11) is smallest prime factor for (2^x)+1 where x=11
2^11+1=2048+1=2049
Prime factors=3 & 683. Samllest prime factor = 3
Hence, g(11)=3

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For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 21:28
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f(x) represents the greatest prime factor of x!
g(x) represents the smallest prime factor of $$2^x$$+1

(g(f(12))?

f(12) = 11 (greatest prime factor of 12!)

(g(11)) = 3 (smallest prime factor of 2^11+1 = 2049 = 3*683)

(g(f(12)) = 3

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 21:37
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f(x) represents the greatest prime factor of x!
g(x) represents the smallest prime factor of $$2^x$$+1
We need to find value of (g(f(12))?
Solution: f(12) equals to greatest prime factor of 12!=12*11*10......*1. Greatest prime factor is 11.
Now, g(11) equals to 2^(11)+1.
2^(11)+1=2049, and smallest prime factor is 3 (B)
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 21:52
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f(12) = greatest prime factor of 12!
The greatest prime factor of 12! is 11
f(12) = 11
g(11) = smallest prime factor of 2^11 + 1
= smallest prime factor of 2049 = 3

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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 22:23
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f(12) = Greatest prime factor of 12! = 11
--> (g(f(12)) = g(11) = smallest prime factor of 2^11 + 1

Note that, 2 cannot be a factor as 2^11 + 1 is ODD

Check for 3,
2^11 + 1 = 2048 + 1 = 2049 --> Divisible by 3

--> g(f(12)) = 3

IMO Option B

Pls Hit Kudos if you like the solution
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For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 22:40
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For every positive integer x, f(x) represents the greatest prime factor of x!, and g(x) represents the smallest prime factor of 2^x+1. What is (g(f(12))?

Here, f(x) represents the greatest prime factor of x!
so, f(12) includes 11 as the greatest prime factor of x!

2^11 +1 = 2049 =3 * 683
So, smallest prime factor is 3
the correct answer choice is (B)
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 22:46
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IMO - B

For every positive integer x,
f(x) represents the greatest prime factor of x!
, and g(x) represents the smallest prime factor of 2^x+1
. What is (g(f(12)) ?

f(12) = 12! => 12 x 11 x 10 x 9 .. and so on.

Greatest prime factor = 11

g(11) = 2^11 + 1 = 992 + 1 = 993

Smallest prime factor of 993 is 3 ( Answer choice B )
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For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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18 Jul 2019, 22:49
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For every positive integer $$x$$,

$$f(x)$$ represents the greatest prime factor of $$x!$$,

$$g(x)$$ represents the smallest prime factor of $$2^x+1$$.

What is $$(g(f(12))$$?

$$f(x)$$=greatest prime factor of x!

$$f(12)$$ = 1*2*3*4*5*6*7*8*9*10*11*12 => greatest prime factor = 11

$$g(x)$$ = smallest prime factor of $$2^x+1$$

$$g(11)$$ = smallest prime factor of $$2^{11}+1$$ => 2048+1 = 2049 = 3*683

Smallest prime factor = 3 => Answer B
For every positive integer x, f(x) represents the greatest prime fact   [#permalink] 18 Jul 2019, 22:49

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