Determine the total number of seven-digit base 10 positive integers divisible by 11 and having the sum of their digits equal to 59. (No computer programs)
ֵ ans: 40
The sum , 59 can be split into 2 parts differing by a multiple of 11 .in one way: 24+35.
Clearly 35 cannot result from summing up the digits on 3 even places within the 7-digit number, it most occupy four.
There is only one partition of 35 into dec. digits ; (8,9,9,9), -S)Making 4 compositions
There are 3 partitions of 24 into dec. digits; 699 (3 compositions), (789 (6 compositions), 888 -1 way onlyl
So the total is 4*(3+6+1)=40.
Smallest number….8699999 .
Rem: a software program to verify and to print the results (sum=59)
should test ONLY numbers running from the 1st to the 2nd , step=11.
Edited on June 7, 2013, 7:34 pm