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

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

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New post 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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New post 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

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

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New post 18 Jul 2019, 12:55
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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

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

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New post 18 Jul 2019, 13:36
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So f(x) = greatest prime factor of x!
g(x)= smallest prime factor of (2^x+1)
Now asked g(f(12)) =?

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

Answer B

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

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New post 18 Jul 2019, 13:48
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Answer B : 3
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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New post 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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New post 18 Jul 2019, 15:27
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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.

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

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New post 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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New post 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 \(g(f(12)) = g(11) = 3.\)

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

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New post 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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New post 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.
The answer is therefore B.

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

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New post 18 Jul 2019, 21:14
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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))?

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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New post 18 Jul 2019, 21:22
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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

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

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New post 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

B is the answer
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New post 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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New post 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


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

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New post 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

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

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New post 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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New post 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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New post 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
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For every positive integer x, f(x) represents the greatest prime fact   [#permalink] 18 Jul 2019, 22:49

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