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Manager  Joined: 25 Feb 2012
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The function p(n) on non-negative integer n is defined in  [#permalink]

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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.

Originally posted by qtrip on 28 Apr 2012, 06:05.
Last edited by Bunuel on 28 Apr 2012, 06:21, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
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Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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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).

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]

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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]

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qtrip wrote:
Thanks Bunuel..I was wondering how something like 11 can be represented. But now I understand why 1024 has to be the right answer.

11 can be expressed as $$p(10,000)=2^0*3^0*5^0*7^0*11^1$$.
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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]

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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).

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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Joined: 10 Oct 2012
Posts: 591
Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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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).

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]

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Hi, Could you use some options in the answer choice & explain the reasoning so that i could understand the concept clearly.

Math Expert V
Joined: 02 Sep 2009
Posts: 58310
Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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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.

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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Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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The case of 1. p(what)=1?
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Joined: 16 Oct 2010
Posts: 9700
Location: Pune, India
Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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zaolupa wrote:
The case of 1. p(what)=1?

If n = 0,
p(0) = 2^0 = 1
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Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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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 question asks for the value that p(n) cannot take.

p(n) is of the form $$2^a * 3^b * 5^c * 7^d$$... etc
We know that this is prime factorization and that every positive integer can be prime factorized. Then what is the constraint on the value of p(n)?
a, b, c, d etc are single digits. So if the prime factorization of a number is $$2^{10}$$ or $$3^{24}$$, it cannot be p(n).

The smallest value that p(n) cannot take is $$2^{10} = 1024$$.

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Re: The function p(n) on non-negative integer n is defined in  [#permalink]

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_________________ Re: The function p(n) on non-negative integer n is defined in   [#permalink] 11 Jul 2019, 07:54
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