Hello, young mathematicians! Today, we’re going on a fun adventure to solve a fascinating math problem. Imagine we have a multi-digit number, let’s call it A. This number is made up of the digits 1, 3, 5, 7, and 9. Each digit can appear more than once, but at least once. The exciting part? This number A can be divided evenly by any of its digits. Our mission is to find the smallest possible value for A. Sounds like a fun challenge, right? Let’s dive in!

First, we know that if a number can be divided by 9, the sum of its digits can also be divided by 9. Our smallest possible number, using each digit at least once, is 13579. The sum of these digits is $1+3+5+7+9=25$. To make this sum divisible by 9, we need to add either 2 or 11 (which is 2+9). Adding 2 means adding two 1s, while adding 11 means adding 119, which adds three more digits. Since we’re looking for the smallest A, we’ll choose to add two 1s. So, our number A is now made up of the digits 1, 1, 1, 3, 5, 7, and 9.

Now, we know that any number can be divided by 1. Also, any number that can be divided by 9 can also be divided by 3. For a number to be divisible by 5, its last digit must be 0 or 5. Since we don’t have a 0 in our digits, the last digit of A must be 5.

So, our problem now is to find the smallest number made up of the digits 1, 1, 1, 3, 5, 7, and 9, ending in 5, that can be divided by 7. A number is divisible by 7 if the alternating sum of every three digits from right to left is divisible by 7.

Let’s list all possible As in ascending order and find the first one that fits this rule:

  • 1113795 -> 795-113+1 = 683, which is not divisible by 7.
  • 1113975 -> 975-113+1 = 863, which is not divisible by 7.
  • 1117395 -> 395-117+1 = 279, which is not divisible by 7.
  • 1117935 -> 935-117+1 = 819, which is divisible by 7. Bingo!

So, the smallest possible value for A is 1117935.

I hope you enjoyed this journey through the world of numbers and divisibility rules. Remember, math is not just about finding the right answers, but also about the fun and exciting journey to get there!