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 D
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The function p(n) on non-negative integer n is defined in
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The function just transform the digits of integer n into the power of primes such as 2, 3, 5, 7, 11 etc.
The smallest positive integer that is not equal to p(n) for any permissible n is the same as finding the least number that cannot be expressed by the function p(n).
Function p(n) is of the form =
$$2^a\ \cdot\ 3^b\ \cdot\ 5^c\ \cdot\ 7^d\cdot\ 11^e\ etc.$$
This is prime factorization and every positive integer can be prime factorized.
The digits of n transform to the power, e.g. function p(n) where n = 9
$$p\ \left(9\right)=\ 2^9$$
Where n = 49
$$p\ \left(49\right)=\ 2^9\ \cdot\ 3^4\ etc$$
Since single digits cannot be more than 10 the function p (n) cannot have the power of 10 or higher.
The least number that cannot be expressed by the function p(n),
$$=\ 2^{10}\ =\ 1,024\ because\ n\ \ne10$$
Option D is CORRECT.
The smallest positive integer that is not equal to p(n) for any permissible n is the same as finding the least number that cannot be expressed by the function p(n).
Function p(n) is of the form =
$$2^a\ \cdot\ 3^b\ \cdot\ 5^c\ \cdot\ 7^d\cdot\ 11^e\ etc.$$
This is prime factorization and every positive integer can be prime factorized.
The digits of n transform to the power, e.g. function p(n) where n = 9
$$p\ \left(9\right)=\ 2^9$$
Where n = 49
$$p\ \left(49\right)=\ 2^9\ \cdot\ 3^4\ etc$$
Since single digits cannot be more than 10 the function p (n) cannot have the power of 10 or higher.
The least number that cannot be expressed by the function p(n),
$$=\ 2^{10}\ =\ 1,024\ because\ n\ \ne10$$
Option D is CORRECT.
