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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, 22:56
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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))?

x = 12
f(12) = Greatest prime factor (12!) = 11

g(11) = Smallest prime factor (2^11 + 1) = (2049) = 3

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, 23:10
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Question: 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 \((g(f(12))\)?

\(f(x) =\) The greatest prime factor of \(12!\)
\(f(12) =\) The greatest prime factor of \(12!\)
\(12! =\text{12 }\times\text{ }\)\(11\) \(\times\text{ 10 }\times\text{9 }\times\text{8 }\times\text{7 }\times\text{6 }\times\text{5 }\times\text{4 }\times\text{3 }\times\text{2 }\times\text{1 }\)
\(11\)\(=\) Greatest Prime Factor

\(g(x) =\) The smallest prime factor of \(2^x+1\)
\(g(\)\(11\)\() =2^{11}+1\)

Since \(2^{11}\) is a little much to calculate without a calculator, start small and look for a pattern.
\[
\begin{vmatrix*}
2^x & + & 1 & = & # & \to & \div 3?\\
2^1 & + & 1 & = & 3 & \to & Yes \\
2^2 & + & 1 & = & 5 & \to & No \\
2^3 & + & 1 & = & 9 & \to & Yes \\
2^4 & + & 1 & = & 17 & \to & No \\
2^5 & + & 1 & = & 33 & \to & Yes
\end{vmatrix*}
\\
\text{Pattern #1: Every answer is odd and therefore not divisible by 2.}\\
\text{Pattern #2: Every odd exponent of 2 with the addition of 1, equals an answer divisible by 3.}\\
\]
\(\text{Since }2^{11} \text{ has an odd exponent, the final answer will be divisible by 3.}\)
\(\text{Therefore, 3 is the smallest prime factor of } (2^{11}+1)\).

Correct Answer: B. 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, 23:10
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We need to find g(f(12)).

Lets break it down.

First lets find out f(12).

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

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

Hence, f(12) = 11.

Therefore, g(f(12)) becomes g(11).

g(11) = smallest prime factor of ((2^11)+1) ie, smallest prime factor of 2049.

Using the options we get 3 as the smallest prime factor of 2049.

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, 23:11
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f(12) = greatest prime factor of 12! = 11

g(11) = 2^11 + 1

It is known that 2^10 = 1024

Then 2^11= 2 * 1024 = 2048

g(11) = 2^11 + 1 = 2049

The sum of all digits is divisible by 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, 23:27
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From the stem we know that:

\(f(x)\) - is the greatest prime factor of \(x!\)
\(g(y)\) - is the smallest prime factor of \(2^y+1\)

Let's figure out step by step what \(g(f(12))\) is. In order to calculate what \(g(y)\) is, we need to first calculate what \(y\) is. Since \(y=f(12)\), we will first calculate what \(f(12)\) is.

1. \(f(12)\) is the greatest prime factor of \(12!\) So \(f(12)=11\) or \(y=11\)

2. \(g(11)\) is the smallest prime factor of \(2^{11}+1\). What is the smallest prime factor of \(2049\) ? \(2049\) is not even, so it can't be \(2\). Is \(2049\) divisible by \(3\) ?
\(2+0+4+9=15\) is divisible by \(3\). Thus \(g(11)=3\)

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 19 Jul 2019, 00:03
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f(12) = factorial of 12 whose greatest prime factor is 11 as 12! is 1*2*3*4*5*6*7*8*9*10*11*12.
Hence, g(f(12)) = g(11) = 2^11 + 1 = 2049 = 3*683. Hence, the smallest prime factor of g(f(12)) is 3.

Correct 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 19 Jul 2019, 00:32
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f(12)= greatest prime factor of 12!. That is 11.
g(f(12))=g(11)= smallest prime factor of \(2^{11}\)+1=2048+1=2049.
Smallest prime number which can divide 2049 is 3.

Hence, ans: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 19 Jul 2019, 00:41
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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)) ?

Given that f(x) = greatest prime factor of x!
g(x) = smallest prime factor of 2^x +1

when x =12 , f(x) = 12! = 12*11*9*8*7*6*5*4*3*2*1
11 is the greatest prime factor
g(11) = 2^11 +1 =2048 +1 = 2049 is divisible by 3
==> 3 is the smallest prime factor

a small Observation here , every odd power of 2 , 2^odd + 1 is a multiple of 3 .

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

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New post 19 Jul 2019, 02:54
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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:
The above function is a compound function, in a compound function, the output of one function becomes the input of second function and so on. while solving this, the first function is the inner most function and the last function is the outermost.
We'll first solve the innermost function.

f(12) = 12! i.e 12 X 11 X 10 X 9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1

We observe the the greatest prime is 11.

We take 11 as the input for the outer most function, it will be g(11)

\(2^{11}\) + 1 = 2048 + 1
=2049.
We can try out the options and divide them by 2049, We see that the smallest prime factor is 3.

Hence 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 19 Jul 2019, 02:56
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\(f(12)\) is the greatest prime factor of 12!.
\(12!=12*11*10*...1\)
The greatest prime factor is 11.
So, \(f(12)=11\)

\(g(f(12))=g(11)\)
\(g(11)\) is the smallest prime factor of \(2^{11}+1\) (=2048+1=2049)
The smallest prime number that can divide 2049 is 3.

Hence, 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 19 Jul 2019, 03:47
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Greatest factor of 12! is 11. 2^11+1 equals to 2049, which is divisible by 3 - smallest prime factor.
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 19 Jul 2019, 04:29
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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(12):
12! = 1*2*3...11*12
The greatest prime factor is 11.
g(11):
2^11+1 = 2049
2049 / 2 = not int
2049 / 3 = int
The smallest prime factor is 3

ANSWER B

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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New post 19 Jul 2019, 05:26
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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(12) = GPF of 12!: GPF of 1*2*3*4*5*6*7*8*9*10*11*12 = 11
\(g(11) = 2^{11}+1 = 2048+1=2049\)

2049 is divisible by 3 as \((2+0+4+9 = 15)\) is divisible by 3. Therefore smallest PF of 2049 is 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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New post 19 Jul 2019, 05:37
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f(12) = highest prime factor of 12 factorial - 11
g(11)=2^11+1=2^10=1024*2+1=2049.
Smallest prime factor is 2, but 2049 cannot be divided by 2, A will not work
B will work 2049 can be evenly divided by 3.
B is the answer
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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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New post 19 Jul 2019, 05:41
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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))?

If f(x) represents the greatest prime factor of x!, the greatest prime factor of it will be 11.
---> (g(f(12))=g(11)=2^x+1=2^11+1=2048+1=2049

it says that g(x) represents the smallest prime factor of 2^x+1 (=2049)
--- > The smallest prime factor of 2049 is 3.

The answer choice 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 19 Jul 2019, 05:57
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. What is (g(f(12))
(g(f(12))?

f(12) = largest prime factor of 12! which is 11
g(11) = smallest PF of 2^11+1 = 2049
so B. 3 is the answer

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

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New post 19 Jul 2019, 06:54
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x=12, f(x) is greatest prime factor of x!, so x! = 12! = 12x11x10.....x1, the greatest prime factor is 11 --> f(x) = 11
g(x) is smallest prime factor of 2^x+1, since x = f(x) = 11 --> g(x) = g(2^11+1)
2^11+1 is odd number (=2049), and the smallest odd prime number is 3, so g(2^11+1) = 3
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 19 Jul 2019, 07:53
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f(12) = Greatest Prime Factor of 12!, that is 11.

g(11)= Smallest Prime Factor of 2^11 + 1
2^11+1=32x32x2+1=2049, which is divisible by 3, so the smallest prime factor is 3.

There might be a better solution to avoid the calculation but this is the best I got.
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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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Re: For every positive integer x, f(x) represents the greatest prime fact  [#permalink]

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New post 18 Jul 2019, 11:12
f(12) = 12!, 11 is the highest prime factor

G(11) = 2^11 + 1, and 2 becomes the lowest prime factor.
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Re: For every positive integer x, f(x) represents the greatest prime fact   [#permalink] 18 Jul 2019, 11:12

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