Determine the respective minimum values of a positive integer N for each p = 1 to 9 inclusively, such that:
- N is composite, and:
- All positive divisors of N (including itself but, with the exclusion of 1) contain the digit p.
Proof that the minimum N for any given p must be semiprime (ie, the product of two primes).
Since N is composite, then it can be expressed as the product of two numbers R and S, neither of which is 1, and both of which include the digit p. Assume that R is composite. All of its factors are also factors of N, so they all (with the exclusion of the number 1) contain the digit p. Thus, R has all the sought-after properties. This is a contradiction, because N was the minimum number which had the sought-after properties. Therefore, our assumption is wrong, and R is prime. Similarly, S must be prime. Therefore, N is semi-prime.
This insight made it significantly easy to find N for any given p. Just multiply small primes which contain the digit p, and N is the minimum product which contains the digit p.
Edited on May 25, 2014, 7:29 pm