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The function p(n) on non-negative integer n is defined in

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The function p(n) on non-negative integer n is defined in [#permalink] New post 28 Apr 2012, 05:05
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The function p(n) on non-negative integer n is defined in the following way: the units digit of n is the exponent of 2 in the prime factorization of p(n), the tens digit is the exponent of 3, and in general, for positive integer k, the digit in the 10^(k–1) th place of n is the exponent on the kth smallest prime (compared to the set of all primes) in the prime factorization of p(n). For instance, p(102) = 20, since 20 = (5^1)(3^0)(2^2). What is the smallest positive integer that is not equal to p(n) for any permissible n?

(A) 1
(B) 29
(C) 31
(D) 1,024
(E) 2,310

OA after some discussion.
[Reveal] Spoiler: OA

Last edited by Bunuel on 28 Apr 2012, 05:21, edited 1 time in total.
Added the OA
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 28 Apr 2012, 05:18
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qtrip wrote:
The function p(n) on non-negative integer n is defined in the following way: the units digit of n is the exponent of 2 in the prime factorization of p(n), the tens digit is the exponent of 3, and in general, for positive integer k, the digit in the 10^(k–1) th place of n is the exponent on the kth smallest prime (compared to the set of all primes) in the prime factorization of p(n). For instance, p(102) = 20, since 20 = (5^1)(3^0)(2^2). What is the smallest positive integer that is not equal to p(n) for any permissible n?

(A) 1
(B) 29
(C) 31
(D) 1,024
(E) 2,310

OA after some discussion.


The function basically transforms the digits of integer n into the power of primes: 2, 3, 5, ...

For example:
p(9)=2^9;
p(49)=2^9*3^4;
p(349)=2^9*3^4*5^3;
p(6349)=2^9*3^4*5^3*7^4;
...

The question asks for the leas number that cannot be expressed by the function p(n).

So, the digits of n transform to the power and since single digit cannot be more than 10 then p(n) cannot have the power of 10 or higher.

So, the least number that cannot be expressed by the function p(n) is 2^{10}=1,024 (n just cannot have 10 as its digit).

Answer: D.

P.S. If you have the OA you have to indicate it under the spoiler.
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 28 Apr 2012, 05:21
Thanks Bunuel..I was wondering how something like 11 can be represented. But now I understand why 1024 has to be the right answer.
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 28 Apr 2012, 05:27
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 25 Jun 2013, 04:56
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 01 Aug 2013, 19:24
Bunuel wrote:
qtrip wrote:
The function p(n) on non-negative integer n is defined in the following way: the units digit of n is the exponent of 2 in the prime factorization of p(n), the tens digit is the exponent of 3, and in general, for positive integer k, the digit in the 10^(k–1) th place of n is the exponent on the kth smallest prime (compared to the set of all primes) in the prime factorization of p(n). For instance, p(102) = 20, since 20 = (5^1)(3^0)(2^2). What is the smallest positive integer that is not equal to p(n) for any permissible n?

(A) 1
(B) 29
(C) 31
(D) 1,024
(E) 2,310

OA after some discussion.


The function basically transforms the digits of integer n into the power of primes: 2, 3, 5, ...

For example:
p(9)=2^9;
p(49)=2^9*3^4;
p(349)=2^9*3^4*5^3;
p(6349)=2^9*3^4*5^3*7^4;
...

The question asks for the leas number that cannot be expressed by the function p(n).

So, the digits of n transform to the power and since single digit cannot be more than 10 then p(n) cannot have the power of 10 or higher.

So, the least number that cannot be expressed by the function p(n) is 2^{10}=1,024 (n just cannot have 10 as its digit).

Answer: D.

P.S. If you have the OA you have to indicate it under the spoiler.




Hi Bunuel,
I am a bit confused here.Cant p(1024) be 2^4 *3^2*5^0*7^1..
kindly elaborate...i get what you mean but am unable to implement it here...
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 02 Aug 2013, 00:22
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up4gmat wrote:
Bunuel wrote:
qtrip wrote:
The function p(n) on non-negative integer n is defined in the following way: the units digit of n is the exponent of 2 in the prime factorization of p(n), the tens digit is the exponent of 3, and in general, for positive integer k, the digit in the 10^(k–1) th place of n is the exponent on the kth smallest prime (compared to the set of all primes) in the prime factorization of p(n). For instance, p(102) = 20, since 20 = (5^1)(3^0)(2^2). What is the smallest positive integer that is not equal to p(n) for any permissible n?

(A) 1
(B) 29
(C) 31
(D) 1,024
(E) 2,310

OA after some discussion.


The function basically transforms the digits of integer n into the power of primes: 2, 3, 5, ...

For example:
p(9)=2^9;
p(49)=2^9*3^4;
p(349)=2^9*3^4*5^3;
p(6349)=2^9*3^4*5^3*7^4;
...

The question asks for the leas number that cannot be expressed by the function p(n).

So, the digits of n transform to the power and since single digit cannot be more than 10 then p(n) cannot have the power of 10 or higher.

So, the least number that cannot be expressed by the function p(n) is 2^{10}=1,024 (n just cannot have 10 as its digit).

Answer: D.

P.S. If you have the OA you have to indicate it under the spoiler.




Hi Bunuel,
I am a bit confused here.Cant p(1024) be 2^4 *3^2*5^0*7^1..
kindly elaborate...i get what you mean but am unable to implement it here...


We don't have to find p(1024). In-fact, the question asks to find the value of the smallest integer which can never be assumed by the function p(n), for any non-negative integer,n.
For eg, for p(n) =5, the initial integer n = 100, for p(n) = 7, n = 1000 and so on. Now, if p(n) were to be 1024 = 2^{10}, that would mean that the units digit of n was 10, which is not possible.

Hope this helps.
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 24 Aug 2014, 12:01
Hi, Could you use some options in the answer choice & explain the reasoning so that i could understand the concept clearly.

thanks in advance
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Re: The function p(n) on non-negative integer n is defined in [#permalink] New post 24 Aug 2014, 12:10
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luckyme17187 wrote:
Hi, Could you use some options in the answer choice & explain the reasoning so that i could understand the concept clearly.

thanks in advance


Solution is given here: the-function-p-n-on-non-negative-integer-n-is-defined-in-131459.html#p1079514 Please read it and also the discussion below it. If something will remain unclear please ask but try to be a little bit more specific. Thank you.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: The function p(n) on non-negative integer n is defined in   [#permalink] 24 Aug 2014, 12:10
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