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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, 08:45
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Well first we have to find F(12)= 2^2*3

The greatest prime factor of f(12) is 3, now g(3)= 2^3 + 1=9

9=3*3

So smallest prime factor is 3

thus g(f(12))= 3
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New post 18 Jul 2019, 08:46
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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))? 

Given, f(12) represents the greatest prime factor of 12!, which is 11, as we know 12! = 12*11*10*9*8*7*6*5*4*3*2*1

Now g(11) represents the smallest prime factor of 2^11+1 = 2048+1 = 2049, which is 3 (2049=3*683, we do not need to calculate all prime factors as we only need the smallest one which is 3)

So, (g(f(12)) = 3
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New post 18 Jul 2019, 08:49
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Greatest prime factor of 12! will be 11. So f(12)=11
g(11)=smallest prime factor of 2^11+1=Smallest prime factor of 2049 which is 3 (2+0+4+9=15 which is divisible by 3 as per divisibility rules)

IMO B
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New post 18 Jul 2019, 08:51
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IMO correct answer is B - 3
explanation as attached
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New post 18 Jul 2019, 08:53
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1) f(12) is the greatest prime factor of 12!, which is 11
2) g(11) is the smallest prime factor of \(2^{11}+1=2049\). not divisible by 2 (the lowest prime) as 2049 is odd, but divisible by 3 (the second-lowest prime) as 2+0+4+9=15, which is divisible by 3

So g(f(12))=3

IMo
Ans: B
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New post 18 Jul 2019, 08:54
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IMO B.

f(x) = largest prime factor of x!
g(x)= smallest prime factor of \(2^x+1\)

g(f(12)), f(12) --> largest prime factor of 12! --> 12X11X10X...1 --> Highest prime factor would be 11.
g(11)= 2^11+1= 2048 + 1 = 2049 .
2049 is not divisible by 2 but is divisble by 3 which is a prime number.

Hence, the answer is 3 which is option B.
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New post 18 Jul 2019, 08:55
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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 2x+1. What is (g(f(12))?

Lets find inner equation value first i.e f(12) = 12! -> What is the greatest prime factor of 12! -> 11
g(11) = \(2^{11}\) + 1 = 2048 +1 = 2049 -> What is the smallest prime factor of 2049
2049 is not divisible by 2
Find out if no if divisible by 3 - add the digits = sum is divisible by 3, hence 2049 is divisible by 3.. smallest prime factor is 3

B is the answer
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New post 18 Jul 2019, 08:57
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Quote:
For every positive integer x, f(x) represents the greatest prime factor of x!, and g(x) represents the smallest prime factor of 2ˆx+1. What is (g(f(12))?

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


f(12) is g8primefac of 12!, which is the largest prime from 0 to 12 = 11;
g(11) is smallest prime factor of 2ˆ(11)+1=2048+1=2049,
notice that 2049 is not divisible by 2, but the sum of its digits is a multiple of 3,
so its divisible by 3, which means the smallest pf is 3.

Answer (B).
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New post 18 Jul 2019, 08:58
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IMO-B

f(x) = Greatest prime factor of x!
g(x) = Smallest prime factor of 2^x + 1


f(12)= Greatest prime factor of 12! = Greatest prime factor ( 12 x 11 x 10 x 9..........x1) = 11
g(11)= Smallest prime factor of 2^11 + 1= Smallest prime factor of 2049 = 3

g(f(12))= g(11)= 3
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New post 18 Jul 2019, 08:59
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Given that f(x) represents the greatest prime factor of x!

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

to find g(f(12))

f(12): since 12! has 11 as the greatest prime factor
therefore, f(12) = 11

Also for g(f(12)): we have (2^11) + 1 = 2049
2049/3 = 683
thus, 3 is the smallest prime factor of g(f(12))

Final answer is 3

Hence 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, 09:04
(g(f(12)) :
F(12) - represent the greatest prime of 12! = 11

g(11) = 2^11+1

The last digit of 2^11 = 4 (based on the pattern of 2,4,8,6)
So the smallest prime or the prime that divides g(11) is 5

IMO : C
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New post 18 Jul 2019, 09:05
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First we have 12!

It can be written as

2^10*3^5*5^2*7*11

Greatest prime factor is 11

Then we have 2^11+1

2049 ,this is not divisible by 11 or 7 or 5 or 2

This is divisible by 3

B it is !!

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New post 18 Jul 2019, 09:06
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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))?


Start by finding f(12)
f(12) = GREATEST prime factor of 12!
12! = 12x11x10x9x8x7x6x5x4x3x2x1

11 is the greatest prime factor

f(12) = 11

g(f(12)) = g(11)

SMALLEST prime factor of \(2^{11} + 1\) = 2048 + 1 = 2049

2049 is not divisible by 2
2049 is divisible by 3.... so 3 is the smallest prime factor of 2049

g(11) = 3 = g(f(12))


Answer: B - 3
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New post 18 Jul 2019, 09:10
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f(x) represents the greatest prime factor of x!
f(12) => 12! = 12*11*10*9*8*7*6*5*4*3*2*1
Thus greatest prime factor is 11.

g(x) represents the smallest prime factor of \(2^x\)+1
g(11) => \(2^{11}\) + 1
=> 2049
2049 = 3*683
Thus smallest prime factor is 3.

IMO the answer is B.

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New post 18 Jul 2019, 09:31
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f(12) is highest prime factor of 12! which is 11.
so g(11) will be lowest prime factor of (2^11+1) => Plowest ( 2049) => 3

Answer shall be B.

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New post 18 Jul 2019, 09:42
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For every positive integer xx, 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

Solution f(12) = highest prime factor for 12! and that would be 11
g(11) = 2^11 +1 = odd number option A out
2^11+1 = 2^10 * 2+1
1024*2 +1 = 2049, which is divisible by 3 hence g(11) wud be the smallest prime factor i.e. 3

Answer = B
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New post 18 Jul 2019, 09:44
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For every positive integer
x
x
,
f(x)
f(x)
represents the greatest prime factor of
x!
x!
, and
g(x)
g(x)
represents the smallest prime factor of
2
x
+1
2x+1
. What is
(g(f(12))
(g(f(12))
?

A. 2
B. 3
C. 5
D. 7
E. 11
Solution:
Its given that for every positive integer x,f(x) represent the greatest prime factor of x!,this implies that from question f(12)=greatest prime factor of 12!
factors of 12!=12*11*10*9*8*7*6*5*4*3*2*1=11*7*\(5^(2)\)*\(3^(5)\)*\(2^(10)\)

As the greatest prime factor is 11 ,hence it implies we need to calculate g(11)=2^(11) +1=2048+1=2049
Now factor of 2049=3*683
Hence g(11)=smallest prime factor of\(2^(11)\) +1=smallest prime factor of 3*683=3
Hence B IMO
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New post 18 Jul 2019, 10:39
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We need to find g(f(12))

f(12), the greatest prime of 12! is 11
Now, g(11), smallest prime factor of 2^11 + 1 = 2048+1 = 2049

Smallest prime number by which 2049 is 3 (using divisibility rule for 3)

B is correct.
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New post 18 Jul 2019, 10:43
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f(12) -greatest prime factor of 12 will be 3.
Therefore question becomes g(3) =2^3+1=9
Smallest prime factor of 9 is 3 so the answer option 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, 11:07
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Given,

1. f(x) => Greatest prime factor of x!
2. g(x) => Smallest prime factor of 2^x + 1

To find,

g(f(12))

Lets do it step by step,

Step 1:

f(12) = Greatest prime factor of 12!

12! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12

=> f(12) = 11

Step 2:

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 2048 + 1 = 2049

Let us check the options which divide 2049

A. 2 --> Wrong. Is not a factor of 2049.

B. 3 --> Correct. Divides 2049 and is the smallest.

C. 5 --> Wrong. Is not a factor of 2049.

D. 7 --> Wrong. Is not a factor of 2049.

E. 11 --> Wrong. Is not a factor of 2049.
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Re: For every positive integer x, f(x) represents the greatest prime fact   [#permalink] 18 Jul 2019, 11:07

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