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Remainder: Part 1 (Posted on 2006-03-20) Difficulty: 3 of 5
A hundred digit number is formed by writing the first few natural numbers in front of each other as follows:

12345678910111213141516171819

Find the remainders when this number is divided by each of the numbers from 1 to 20.

No Solution Yet Submitted by Ravi Raja    
Rating: 2.5000 (2 votes)

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Hints/Tips A start (partial solution) | Comment 1 of 6

Well, ok, here's what I could do with a pen and an 8-digit pocket calculator:

The hundred digit number will end with ...5253545 (right in the middle of 55).

divisor remainder reason
------- --------- ------
   1        0     obvious
   2        1     number ends in 5
   3        2     digits sum to 5
   4        1     last two digits = 45
   5        0     number ends in 5
   6        5     remainder when divided by 2 = 1, and by 3 = 2;
   
   8        1     last three digits = 545
   9        5     digits sum to 5
  10        5     number ends in 5
  11        1     sum of 1st, 3rd, etc digits - sum of 2nd, 4th, etc.
  12        5     remainder when divided by 3 = 2, and by 4 = 1
 
  15        5     ends in 5, remainder when divided by 3 = 2
  16        9     last four digits = 3545
  18        5     remainder when divided by 2 = 1, by 9 = 5
 
  20        5     last two digits = 45
 
 

  Posted by tomarken on 2006-03-20 11:12:33
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