We are given a problem to find the difference between the largest and smallest five-digit integers with specific properties:

1. Each digit of the number must be 1, 2, or 3.
2. Each of the digits 1, 2, and 3 must appear at least once.
3. The number is not divisible by 2 or by 3.

Step-by-Step Breakdown:

Step 1: Divisibility Conditions

• A number is divisible by 2 if its last digit is even, i.e., if the last digit is 2. Hence, the number cannot end with 2.
• A number is divisible by 3 if the sum of its digits is divisible by 3. To ensure divisibility by 3 doesn't occur, we will calculate the sum of digits and check for non-divisibility.

Step 2: Constructing the Largest and Smallest Numbers

To maximize or minimize the five-digit number while satisfying all conditions, we will ensure that:

• The digits are from the set {1, 2, 3}.
• The sum of the digits is not divisible by 3.
• The last digit is not 2, to avoid divisibility by 2.

Constructing the Largest Number:

To form the largest number:

• We should start with the largest possible digits first (i.e., 3).
• The largest candidate with each digit appearing at least once, starting with 3's, would be: 33321 (with the last digit not being 2 to avoid divisibility by 2).

We check if this number is divisible by 3:

• Sum of digits = $3 + 3 + 3 + 2 + 1 = 12$, which is divisible by 3.
• Hence, 33321 is not valid.

Next, we modify the number to avoid divisibility by 3, by reducing one of the digits:

• Let's try 33312:
  • Sum of digits = $3 + 3 + 3 + 1 + 2 = 12$, still divisible by 3.

Finally, try 33123:

• Sum of digits = $3 + 3 + 1 + 2 + 3 = 12$, still divisible by 3.

After adjusting and checking multiple combinations, the correct largest valid number can be 32313 which satisfies all conditions.

Constructing the Smallest Number:

For the smallest number:

• We should start with the smallest possible digits first (i.e., 1).
• The smallest candidate with each digit appearing at least once, starting with 1's, would be: 11233.

We check if this number is divisible by 3:

• Sum of digits = $1 + 1 + 2 + 3 + 3 = 10$, which is not divisible by 3.

Also, the last digit is not 2, so it is not divisible by 2.

Hence, the smallest valid number is 11233.

Step 3: Finding the Difference

Now, the largest number is 32313 and the smallest number is 11233.

The difference between them is: $32313 - 11233 = 21080$

Thus, the difference between the largest and smallest numbers is 21080.

Would you like further details or explanations on any of the steps? Here are five related questions that build on this concept:

1. How do divisibility rules affect the construction of numbers in general?
2. Can you generalize this problem for six-digit numbers?
3. What would be the effect if we allowed repetition of some digits more than once?
4. How can we construct numbers under similar constraints but allow divisibility by 2 or 3?
5. What would change if one of the digits (1, 2, or 3) were excluded entirely?

Tip: When working with divisibility rules, remember that small changes in the sum of digits can help avoid divisibility by 3 while maintaining other constraints.