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Eight Embodiment (Posted on 2014-05-25) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

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Key Insight | Comment 5 of 6 |
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
  Posted by Steve Herman on 2014-05-25 19:26:03

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