For a positive base ten integer of the form ABCD drawn at random between 1111 and 9999 inclusively, determine the probability that ABCD is divisible by each of AB, AC, AD, BD and CD; but, not divisible by BC.

Note: Each of the letters in bold represents a digit from 1 to 9, whether same or different.

There are 6561 four digit numbers using only digits 1..9.  Of these three are divisible by all five (AB,AC,AD,BD,CD) while not being divisible by BC:

1995 div by 19 19 15 95 95 not div by 99

2184 div by 21 28 24 14 84 not div by 18

3774 div by 37 37 34 74 74 not div by 77

The probability is 3 / 6561 or c. 0.000457

edited for typos

Edited on October 19, 2010, 6:03 pm

Edited on October 19, 2010, 6:06 pm

Posted by ed bottemiller on 2010-10-19 17:54:29