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Divisibility and Digit Sum (Posted on 2013-06-07) Difficulty: 3 of 5
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)

No Solution Yet Submitted by K Sengupta    
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Solution No Subject | Comment 2 of 8 |

                                                                                                                                                      ֵ ans: 40

Sol:
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   .

Biggest…………………9999968     

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
  Posted by Ady TZIDON on 2013-06-07 11:51:36

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