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18-yes, 17-maybe (Posted on 2010-08-06) Difficulty: 2 of 5
Prove that among any 18 consecutive 3-digit numbers there must be at least one which is divisible by the sum of its digits.

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Solution solution | Comment 3 of 4 |
case 1)max sum of digits of 3-digit number = 27
when number is 999
Here 18 consecutive digits are 982,983,...,999.
In this case 990 is divisible by 18.

case 2)In 18 consecutive 2 are divisible by 9.
one is an even multiple and other is odd multiple.

rule of divisibility by 9: divisible by sum of digits.
So, sum of digits can be 9,18.

The even multiple of 9 is divisible by both 9 and 18.

Hence Proved.

  Posted by Praneeth on 2010-08-06 19:00:40
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