(I) N is a duodecimal (base 12) 50 digit positive integer of the form XX....Y, where X is repeated precisely 49 times followed by Y, such that N is divisible by the duodecimal number 137. Each of X and Y is a different duodecimal digit from 1 to B.

Determine the possible value(s) of N.

(II) P is a duodecimal (base 12) 50 digit positive integer of the form XX....Y, where X is repeated precisely 49 times followed by Y, such that P is divisible by the duodecimal number 147. Each of X and Y is a different duodecimal digit from 1 to B.

Determine the possible value(s) of P.

A 49-digit duodecimal number XX...XX would be X*(12^49-1)/11 (written in decimal). Therefore the 50-digit XX...XXY is 12*X*(12^49-1)/11 + Y. We want this to be divisible by the duodecimal number 137, which is the decimal number 144+36+7=187.

list
   10   for X=1 to 11:for Y=0 to 11
   20     N=12*X*(12^49-1)//11+Y
   30     if N @ 187=0 then print X;Y
   40   next:next
OK
run
 5  8
OK

So N consists of 49 duodecimal 5's and an 8.

The duodecimal number 147 is the decimal number 144+48+7=199.

list
  110   for X=1 to 11:for Y=0 to 11
  120     P=12*X*(12^49-1)//11+Y
  130     if P @ 199=0 then print X;Y
  140   next:next
OK
run
 9  5
OK

So P consists of 49 duodecimal 9's and a 5.

BTW, in case you're interested, the decimal values of N and P are 413656279545552262423981148879207560223961481966208468 and 744581303181994072363166067982573608403130667539175233 respectively.

Posted by Charlie on 2010-11-06 18:31:52