Charmaine has written down three 3-digit positive integers which between them contains exactly 9 distinct digits.
Among these three positive integers:
- There is at least one that is divisible by 2.
- There is at least one that is divisible by 3.
- There is at least one that is divisible by 4.
- There is at least one that is divisible by 5.
- There is at least one that is divisible by 6.
- There is at least one that is divisible by 7.
- There is at least one that is divisible by 8.
- There is at least one that is divisible by 9.
In addition, it is known that all the three positive integers are divisible by 11.
What are the three 3-digit integers posited by Charmaine?
Note: Adapted from Enigma # 1776 which appeared in 'New Scientist' in 2013.
(In reply to
Answer by Brian Smith)
Brian:
Your first statement is not correct:
"A three digit number that is divisible by 11 either has its middle digit equal to the sum of the other two, or the middle digit is zero and the other two digits sum to 11."
Other possibilities are:
middle digit Sum of first and last
------------- -----------------------
1 12
2 13
3 14
4 15
5 16
6 17
By the way, I am a big fan. I really appreciate that you submit analytical solutions. Computers might be fast sometimes, but they obscure the math aspects of the solution.
re: Answer
