Digits

My approach was not very graceful, I'll admit, but I wanted to test all reasonable possibilites, because the problem was flagged as having multiple solutions. I found that there are not multiple solutions. I think the flag should come off.

First, using the divisibility rules for 5, and since 0 is not available, the 5th digit must be 5.

Now to fill the other 8 positions. Using the multiplication facts and pattern recognition, if a number is divisible by 2, 4, 6, or 8, then its last number must be a 2, 4, 6, or 8. Therefore the 2nd, 4th, 6th, and 8th digits must be 2, 4, 6, or 8. This narrows down the possibilities, leaving the 1st, 3rd, 7th, and 9th position to be 1, 3, 7, or 9.

Using some Combinatorics: To choose 4 things from a list of 4 choices, without replacement, there are 4x3x2x1 = 24 ways. So there are 24 ways to choose the 4 even numbers and 24 ways to choose the four odd numbers. These are independent events, so 24x24=576 ways to fill the other 8 positions.

So when I realized that 576 possibilities are a lot fewer than the 9! = 362880 ways of filling 9 spaces from 9 choices, I thought analyzing 576 items can be done in a spreadsheet. I listed all of the 576 possible combinations and checked each combination for divisibility in each of the 9 strings a, ab, abc, abcd, abcde, abcdef, abcdefg, abcdefgh, and abcdefghi. If a string had a remainder when divided, it was tagged.

Amazingly, there was only ONE combination that had no tags and popped out as solving the problem: 381654729

Sue

Its solution is 381654729 . You must known the divisibility rules of 1 to 9. Then see my solution ...............

At first , I make which 3 digits number is divisible by 8. 896, 872, 832, 816, 672, 632, 496, 472, 432, 416, 296, 216 etc.

( Its must be 2nd, 4th, 6th, 8th position digit is even . So, Its must be 7th position digit is odd. )

So, 6th , 7th, 8th digits number are one of the above.

Now , 5th digit must be 5.

Test sample 1:

So, At first we test , 000058960 . ( Other digit are 1, 2, 3, 4 , 7 ). ( Here , 0 digit means we do not khow which digit are there. )

Now , at first we check which couple are divisible by 4. We get 001258960, 003258960, 007258960. 2nd position must be even. So, we get , 041258960 , 043258960 , 047258960 .

Now, we need which number is divisible by 3.

So , we get , 741258960 , 147258960 So, Our value are 741258963, 147258963. At last you need check 7412589 , 1472589 are divisible by 7 or not. But two number are not divisible by 7. Which number is divisible by 7 is the answer. Now, You check , 000058720 , 000058320, 000058160 , 000056720, 000056320, 000054960 , 000054720 , 000054320 , 000054160 , 000052960 , 000052160 .

You must get your solution . I am weak is english. So, forgive me.