In the following statements, each of the capital letters in bold represent a different decimal digit from 1 to 9.

COUSCOUS is an 8-digit positive integer none of whose digits is composite and it is divisible without remainder by each of the primes CU, S, F, CFU and UF but not by the prime number CC.

What is the number represented by FOCUS?

FOCUS = 32175, as follows:

C must obviously be 1, since CC = 11 is the only such double-digit prime.  Since 9 as a composite number is not in COUSCOUS, that leaves 13 or 17 as the only remaining two-digit possibilities for CU.  Accordingly, COUSCOUS is shaping up to look like 1_3_1_3_ or 1_7_1_7_, and CFU will look like 1_3 or 1_7.  Excluding those primes with a zero, CFU can only be 163, 173, 193, or 127, 137, 157, 167, 197.  Reversing the last two digits of these choices reveal the only possible primes for UF are 37, 73 or 79.

With 1 being C, 3 and 7 the possible choices for U, and 3, 7 or 9 the possible choices for F, 2 and 5 remain as the likely prime choices for O and S.  Four possible arrangments for COUSCOUS emerge:  12351235, 15321532, 12751275 and 15721572.  None of these are divisible by 9 as a possibility for F, so U and F must be 3 and 7 in some order.

For the first two options (with U = 3), CU =13, CFU therefore = 173, F = 7, UF = 37, etc., all of the required divisions cannot be carried out without some remainders.

Therefore for the second pair of options with U = 7, CU = 17, CFU = 137, F = 3, UF = 73, etc., all of the required divisions can only be completed with 12751275, having O =2 and S = 5.

Hence, FOCUS = 32175  

   

Posted by rod hines on 2008-11-10 18:59:52