Smallest prime that does not divide any 5-digit number

 
 
Find the smallest prime that does not divide any 5-digit number whose digits are in a strictly increasing order.

 

Solution :

Let digit number   n   be a 5-digit number

n \; = \; abcde

with    0 \; < \; a \; < \; b \; < c \; < \; d \; < \; e \; < \; 10

Let   S \; = \; (a+c+e) - (b+d)

Then

S \; = \; a + (c-b) + (e-d) \; > \; a \; > \; 0

and

S \; = \; e - (d-c) - (b-a) \; < e \; \leq \; 10

so   S   is not divisible by 11 and hence   n   is not divisible by 11,

Thus 11 is the smallest prime that does not divide any 5-digit number whose digits are in a strictly increasing order.

 
 
 
 
 
 

 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Prime Numbers and tagged , . Bookmark the permalink.

2 Responses to Smallest prime that does not divide any 5-digit number

  1. paul says:

    I recon its 11, the next one after that is 97.

    here is a bit of MMA code

    c=1;n=1;
    nums=FromDigits/@Cases[IntegerDigits/@Range[10000,99999],{a_,b_,c_,d_,e_}/;a<b<c<d<e];
    While[c!=0,c=Count[Mod[nums,Prime[n]],0];n++];
    Print[Prime[n-1]]
    

    Paul.

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