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) the 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
The OA is D.
Please, can any expert explain this PS question for me? I tried to solve it but I can't get the correct answer and I would like to know how to solve it in less than 2 minutes. I need your help. Thanks.
The function p(n) on non-negative integer n is...
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The key here is to see that so long as no prime base is raised to an exponent greater than 9, that number could be expressed as p(n). 2^3 * 3^4 * 5*7, for example would be p(743), right? But there'd be no way to express, say 2^11, as each digit in n represents an exponent for one prime base. Put another way, if the exponent in question isn't a single digit, there's no way to express it in this framework. So now we just take the prime factorization of the answer choices until we find one that has a prime base raised to an exponent that cannot be expressed as a single digit.swerve 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) the 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
The OA is D.
Please, can any expert explain this PS question for me? I tried to solve it but I can't get the correct answer and I would like to know how to solve it in less than 2 minutes. I need your help. Thanks.
A) 1 = 2^0, 0 is a single digit. So this could be expressed as p(n)
B) 29 = 29^1; 1 is a single digit. So this could be expressed as p(n)
C) 31 = 31^1; 1 is a single digit. So this could be expressed as p(n)
D) 1024 = 2^10. 10 is a two digit number, so there's no way to depict this value in the given framework. The correct answer is D