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

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))?

Largest prime factor of f(12)=11
Now g(11)=2^11+1

Now, we know with power of 2...units digit can be 2,4,8,6
For 2^11, units digit would be 8.
2^3=8 and 8+1=9 which is divisible by 3
Similarly, 2^7=128 & 128+1=129 (again divisible by 3)

Hence based on pattern 2^11 +1 would be divisible by 3

Hence B

f(12) = largest prime divisor of 12! ==> 11

Now g(11) = smallest prime factor of 2^11 +1
2 cannot be the factor.

Try with 3
Remainder
(-1)^11 +1
-1+1=0
3 is smallest prime factor.

Hence, B is the answer.

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

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

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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