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

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.

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.

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