IMC 2007 Q19
19. The following sequence continues indefinitely:
Which of the following integers is a multiple of 81?
- A 200 007
- B 20 000 007
- C 2 000 000 007
- D 200 000 000 007
- E 20 000 000 000 007
Intermediate Mathematical Challenge 2007, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 20 000 000 000 007
As each of the five given options is a member of the sequence, each is a multiple of 9. So we require the third factor of the number, that is the factor consisting of several 2s followed by a single 3, to be a multiple of 9 also. This is true if and only if the sum of its digits is a multiple of 9. For each number in the sequence, the number of 2s in its third factor equals the number of 0s in that number. The options given have 4, 6, 8, 10, 12 zeros, corresponding to their third factors having digital sums of 11, 15, 19, 23, 27 respectively. Of these, only 27 is a multiple of 9 so the correct answer is 20 000 000 000 007.
Solution © UK Mathematics Trust, from the IMC 2007 solutions.