The missing digit is 4.

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.

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

The number 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.