Question

Determine the smallest prime that does not divide any five–digit number whose digits are in a strictly increasing order.

Answer 1

Note that 12346 is even, 3 and 5 divide 12345, and 7 divides 12348. 

Consider 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 ≤ 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 fivedigit number whose digits are in a strictly increasing order.