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

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

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18 Jul 2019, 07:59
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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

 This question was provided by Veritas Prep for the Game of Timers Competition

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

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Updated on: 21 Jul 2019, 21:22
1
Given:
f(x): greatest prime factor of x!
g(x): smallest prime factor of $$2^x + 1$$

f(12): greatest prime factor of 12! = 12*11*10*9*8*7*6*5*4*3*2*1 -> 11
g(11): smallest prime factor of $$2^{11} + 1$$= 2049 = 3 * 683 -> 3

Therefore, (g(f(23)) = 3

Originally posted by Sayon on 18 Jul 2019, 08:08.
Last edited by Sayon on 21 Jul 2019, 21:22, edited 2 times in total.
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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, 08:10
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$$f(12)$$ is the greatest prime factor of $$12!$$ which is $$11$$

$$g(11)$$ is the smallest prime factorial of $$2^11+1$$

$$2^11+1 = 2049$$

$$2049$$ is an odd number and the sum of digits of $$2049$$ is $$15$$ which is a multiple of $$3$$. So $$2049$$ is divisible by $$3$$ and so $$3$$ is the smallest prime factor of $$2049$$

$$g(f(12)) = 3$$

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

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Updated on: 19 Jul 2019, 03:22
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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: 1. For every positive integer x, f(x) represents the greatest prime factor of x!
2. For every positive integer x,g(x) represents the smallest prime factor of 2^x+1

12!=1*2*3*4*5*6*7*8*9*10*11*12=> Prime factors = 2,3,5,7 and 11 => Greatest prime factor = 11
f(12) = greatest prime factor of 12! = 11
g(f(12)) = g(11) = smallest prime factor of (2^11+1 = 2048+1= 2049)
Since 2049 is odd, 2 is not a factor
2049/3 = 643, 3 is a factor of 2049
g(11) = 3 = g(f(12))

IMO B
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Originally posted by Kinshook on 18 Jul 2019, 08:10.
Last edited by Kinshook on 19 Jul 2019, 03:22, edited 2 times in total.
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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, 08:12
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f(12) gives 12! so greatest prime factor is 11
g(11) gives us (2^11 + 1), i.e. 2049, smallest prime factor of which 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, 08:14
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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))?

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
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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, 08:22
1

largest prime factor of f(12)= 12! is 11
and g(11) = 2048 = 683*3
as 683 is odd, smaller prime number than 3, which is 2 is not possible, so answer 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, 08:28
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B

First off, we need to find the greatest prime factor of 12!. So, 12! = 12*11*10 ....*1

Clearly, 12 is not prime, so greatest prime factor is 11. So f(12) = 11

Now, g(f(12)) = g(11) = 2^11 + 1 = 2049

Factorize 2049, we get 2049 = 3*683

We dont have to try to factorize 683 further since 3 is the smallest prime factor possible.
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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, 08:29
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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.

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

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Updated on: 18 Jul 2019, 19:34
1
First, we have to find f(12), which is greatest factor of 12!, so we have 11 from this one.

Then, we have to find out g(11) = 2^11 +1, which is 2048 + 1 = 2049, factorizing, we have that it has only two factors 3 and 683.

g(f(12))= 3, so (B) is our answer

Originally posted by Mizar18 on 18 Jul 2019, 08:31.
Last edited by Mizar18 on 18 Jul 2019, 19:34, edited 2 times in total.
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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, 08:35
1
Quote:
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)= greatest prime factor of 12 !
As 12! = 12 *11* 10!
therefore f(12) = 11
Now g(11)= smallest prime factor of $$2^(11)$$ +1

$$2^(10)$$ = 1024
$$2^(11)$$ = 2048
therefore $$2^(11)$$ +1 = 2049
divisible by 3
Hence option 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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18 Jul 2019, 08:38
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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))?

This is an easy question.
F(12) = 12! so the greatest prime factor is going to be 11.
g(11) = 2^11 +1 = 2048+1 = 2049. (Trick for divisibility the total is divisible by 3 and not be 2)
so the smallest prime factor of G(11) is going to be 3.

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

Hence the answer is B. (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, 08:40
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

f(12) = greatest prime factor of x! = 11 (Because 12! = 12*11*10*9*...*1)
g(11) = smallest prime factor of $$2^{11}$$+1
$$2^{11}$$= 2048

So, we need to find the smallest prime factor of 2049.
Can't be divided by 2. But the sum of digits is divided by 3 so, Ans should be (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, 08:40
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The greatest prime factor of 12! is 11.
Ok, the next step is 2^(11)+1=2049
2049 is not divisible by 2 because it is odd
But it is divisible by 3
And thus 3 is the answer

IMO 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, 08:40
1
IMO :B

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

Sol:

greatest prime factor of 12! is 11, because rest of the number will be smaller than 11 and the numbers greater than this number will not be prime numbers.
then g((2^11)+1)=2048+1=2049

2049 is divisible by 1,3... 2049 so 3 is the smallest prime factor for g(x)

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

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Updated on: 18 Jul 2019, 08:43
1
Quote:
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 $$(g(f(12))$$?

The task tells us that $$f(x)$$ represents greatest prime factor of $$x!$$. Prime factor is a positive integer which is not equal to $$1$$ and has two factors: itself and $$1$$. Let us find $$f(x)$$ first, and then find $$g(f(x))$$.
$$f(12)$$ = greatest prime factor of $$12!$$
Let us find out what greatest prime factor of $$12!$$ is: $$12! = 1 * 2* 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12$$.
As we can see from above, the greatest of the numbers is $$11$$ which is the greatest prime factor for $$12!$$
$$12!$$ cannot have higher prime factor since $$12$$ itself is equal to $$2*2*3$$, and any product of its numbers will have more factors than $$2$$ and will not be prime.
Hence, $$f(12) = 11$$

Now, let us find what $$g(f(12))$$ is.
$$g(f(12)) = g(11)$$ and is the smallest prime factor of $$2^1+1$$
$$2^1+1=2048+1=2049$$
Let us find what are the factors for $$2049$$. As $$2+0+4+9=15$$ and is divisible by $$3$$, $$3$$ is one of prime factors for $$2049$$. Also, we can notice that $$2049$$ is odd, and hence not dividable by $$2$$.
Because of this, the smallest prime factor of $$2049$$ is $$3$$.
$$g(11) = 3$$.

Originally posted by RusskiyLev on 18 Jul 2019, 08:41.
Last edited by RusskiyLev on 18 Jul 2019, 08:43, edited 1 time in total.
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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, 08:41
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Lets break down the simplification of g(f(12)) in to two steps.

1. f(12) - f(12) will be the largest prime factor of 12! (as stated in the question). $$12!=12*11*10*9*8*7*...*2*1$$. It is obvious that the largest prime factor here will be 11. Lets move to the next step now.

2. $$g(f(12))$$ - $$f(12)=11$$, so expression becomes, $$g(11)$$, $$g(11)$$ is defined as the smallest prime factor of 2^(11)+1, now as 1 is being added to a power of 2 the resulting value shall be odd, so 2 can't be the smallest prime factor. 2^(11)=2048+1=2049. The next smallest prime factor after 2 is 3, we apply the divisibility by 3 rule to see if 3 is a factor of 2049. Divisibility rule of 3 states that if the sum of the digits of any number is divisible by 3 then that number is also divisible by 3. $$2+0+4+9=15$$ and 15 is divisible by 3 so 2049 is thus also divisible by 3. You can also obviously do the division directly without knowing the divisibility rule to see if $$\frac{2049}{3}$$ is an integer and you will see that it is and that $$\frac{2049}{3}=683$$ but knowing the rule saves us the extra step and speeds things up.

$$g(f(12))=g(11)=3$$. 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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18 Jul 2019, 08:42
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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))?
Solution:
f(12) = greatest prime factor of 12! = greatest prime factor of (12*11*10!) = 11
g(f(12) = g(11) = smallest prime factor of $$2^11+1$$ = smallest prime factor of 2049 = smallest prime factor of (3* 683) = 3. (2^11 = 1024*2=2048)

A. 2
B. 3 --> correct
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, 08:44
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f(12) => 12*11*10....1
greatest prime=11
g(11)= 2^11+1=2049
Smallest prime factor of 2049
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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, 08:45
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f(x)= greatest prime factor of x!
=> f(12)= greatest prime factor of 12! = 12*11*10*...
f(12)=11

g(11)=smallest prime factor of 2^11+1
=> smallest prime factor of 2049
Start checking from smallest prime number we get 3 which is a factor of 2049.
Hence, g(f(12))=3. Option B
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Re: For every positive integer x, f(x) represents the greatest prime fact   [#permalink] 18 Jul 2019, 08:45

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