Calculate the positive square root of 100 using the prime factor method, and arrange the following numbers in descending order: 69831, 69813, 98813, 96831, 86931.

Here are the solutions to your problems:

1. Positive square root using prime factor method

Step 1: Identify all prime numbers greater than 1 and less than 25.
The prime numbers are $2, 3, 5, 7, 11, 13, 17, 19, 23$.

Step 2: Calculate the sum of these prime numbers.

Step 3: Find the prime factorization of the sum (100).

Step 4: Obtain the positive square root using the prime factors.

The positive square root is $\boxed{10}$.

2. Arrange numbers in descending order

Step 1: List the given numbers: $69831, 69813, 98813, 96831, 86931$.

Step 2: Compare the numbers starting from the leftmost digit (tens of thousands place).
The largest digit in the tens of thousands place is 9. The numbers starting with 9 are 98813 and 96831.
Comparing 98813 and 96831: $98813 > 96831$.
The next largest digit in the tens of thousands place is 8. The number is 86931.
The next largest digit in the tens of thousands place is 6. The numbers starting with 6 are 69831 and 69813.
Comparing 69831 and 69813: $69831 > 69813$.

Step 3: Arrange them in descending order (largest to smallest).

The numbers in descending order are $\boxed{98813, 96831, 86931, 69831, 69813}$.

3. Evaluate fractions

Step 1: Find the least common multiple (LCM) of the denominators $2, 3, 4, 5$.
The LCM of $2, 3, 4, 5$ is 60.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 60.

Step 3: Perform the operations.

The answer in improper fraction form is $\boxed{\frac{13}{60}}$.

4. Value of expression and place value

Step 1: Perform the subtraction $4986093 - 948699$.

$----------$

Step 2: Perform the division $4037394 \div 125$.
Using long division:

Since the question implies an integer division context (and place value of digit 1), let's re-check the problem statement. "Find without using a calculator, the value of (4986093-948699) $\div$ 125". This implies an exact division or a result where the place value of 1 is clear. Let's assume the question expects an integer result, which means there might be a typo in the numbers or the division is not exact. However, if we perform the division as given:

So, $4037394 \div 125 = 32299$ with a remainder of 19.
If the question expects an exact value, it would be $32299 \frac{19}{125}$ or 32299.152.
Given the instruction "without using a calculator" and "place value of digit 1", it's likely an integer answer is expected, or the question implies a specific context for the result. Let's assume the result is 32299 (integer part).

If we consider the exact decimal value: 32299.152. The digit '1' is in the tenths place.
If we consider the integer part 32299, there is no digit '1'. This suggests there might be an error in my interpretation or the problem itself.

Let's re-evaluate the subtraction and division carefully.
$4986093 - 948699 = 4037394$.
$4037394 \div 125$.
Let's perform long division:

        32299
    _______
125 | 4037394
      -375
      ----
       287
      -250
      ----
        373
       -250
       ----
        1239
       -1125
       -----
         1144
        -1125
        -----
           19

So, $4037394 \div 125 = 32299$ with a remainder of 19.
The question asks for "the value of" and then "place value of digit 1 in the answer obtained above". This implies the answer should contain a '1'.

Let's consider if the question implies rounding or if there's a specific context for the answer. If the answer is 32299.152, then the digit '1' is in the tenths place. If the question implies an integer answer, then there is no '1'.

Given the context of "without using a calculator", it's common for such problems to have exact integer answers. Let's re-check the numbers for any common factors or if the numbers are designed to give a specific result.
$125 = 5^3$. For a number to be exactly divisible by 125, its last three digits must be divisible by 125.
394 is not divisible by 125. So the division is not exact.

If the question expects an integer answer, and there is no '1' in 32299, then the question about the place value of '1' is problematic.
However, if we consider the decimal representation 32299.152:
The digit '1' is in the first decimal place, which is the tenths place.

Let's assume the question expects the decimal value and then asks for the place value of '1'.
The value is $\boxed{32299.152}$.
The digit '1' in 32299.152 is in the tenths place.

5. Consecutive odd integers