There is a 10-digit multiple of 234. 9 of its digits in ascending order are: 0, 1, 1, 2, 3, 4, 5, 7 and 9. What is the missing digit?

Saturday, May 14, 2011

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### 2 comments

### Credit

This is an original puzzle from cotpi.

## Vikram Agrawal solved this puzzle:

4 is the missing digit.

There are 25519 combinations of 0, 1, 1, 2, 3, 4, 4, 5, 7 and 9 which are 10-digit multiple of 234. Interestingly, for the given 9 digits 0, 1, 1, 2, 3, 4, 5, 7 and 9, no digit other than 4 gives a 10 digit combination that is multiple of 234.

## Susam Pal from cotpi added:

4 is the correct answer and indeed there are 25519 10-digit multiples of 234 which have 0, 1, 1, 2, 3, 4, 5, 7 and 9 as 9 of its digits. The remaining digit must be 4. We need not brute force to confirm this. It can be proven logically.

234 is a multiple of 9. So any multiple of 234 is also a multiple of 9. We know that the sum of all digits of a multiple of 9 is a multiple of 9. If we assume the remaining digit to be x, the sum of the digits of the multiple in question is 0 + 1 + 1 + 2 + 3 + 4 + 5 + 7 + 9 + x = 32 + x. The only value of digit x that makes 32 + x a multiple of 9 is 4.