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.
Well, ok, here's what I could do with a pen and an 8digit 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 20060320 11:12:33 