Let's carefully step through the problem to find this special 5-digit number.
1. Range:
- The number is between 34500 and 46000.
2. Number is even:
- The unit (ones) place of the number must be 0, 2, 4, 6, or 8.
3. Multiple of 5:
- The unit (ones) place of the number must be 0 or 5. Given the number is even, the unit (ones) place must be 0.
4. Thousands and ones place:
- The digit in the thousands place is repeated in the ones place.
- Therefore, the ones place is 0, hence the thousands place should also be 0.
5. Sum of digits:
- The sum of all five digits must be 16.
6. Biggest one digit even number in tens place:
- The biggest single-digit even number is 8, so the tens place must be 8.
7. Tens and hundreds place are same:
- Since the tens place is 8, the hundreds place must also be 8.
From the provided conditions, the structure of the number can be visualized as:
[tex]\[ 34X88 \][/tex]
Let's break down the digits and find their sum to satisfy the condition that the sum of the digits should equal to 16:
[tex]\[ 3 + 4 + X + 8 + 8 = 16 \][/tex]
Now, we simplify the equation:
[tex]\[ 3 + 4 + X + 16 = 16 \][/tex]
[tex]\[ 7 + X + 16 = 16 \][/tex]
[tex]\[ X + 7 + 16 = 16 \][/tex]
[tex]\[ X + 7 = 0 \][/tex]
Clearly, solving this equation provides us:
[tex]\[ X = 0 \][/tex]
However, X being 0 doesn't fit within the original range (34500 to 46000). Let’s reevaluate or try another approach.
Thus, we see that under the conditions provided, there is no number that meets every single criterion, resulting in no valid 5-digit number within the given constraints. Hence, there is no such number satisfying all the given conditions.
