The sum of the factors of 9! Which are odd and are of the form `3m+2`, where m is a natural number is `8p+q`, then the value of `|p-q|` is __________

Answer

To solve the problem, we need to find the sum of the odd factors of \(9!\) that are of the form \(3m + 2\), where \(m\) is a natural number. Then, we will express this sum in the form \(8p + q\) and find the value of \(|p - q|\).

### Step 1: Calculate \(9!\)

First, we calculate \(9!\):
\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 2: Prime Factorization of \(9!\)

Next, we find the prime factorization of \(9!\):
\[ 9! = 3^4 \times 2^7 \times 5^1 \times 7^1 \]

### Step 3: Identify Odd Factors

To find the odd factors of \(9!\), we ignore the factor of \(2\) since we want odd factors. The odd part of \(9!\) is:
\[ 3^4 \times 5^1 \times 7^1 \]

### Step 4: Determine the Form \(3m + 2\)

We need to find the odd factors of the form \(3m + 2\). The odd factors can be expressed as:
\[ 3^a \times 5^b \times 7^c \]
where \(0 \leq a \leq 4\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\).

### Step 5: List Possible Odd Factors

Now, we list the odd factors:
- For \(a = 0\): \(5^0 \times 7^0 = 1\), \(5^0 \times 7^1 = 7\), \(5^1 \times 7^0 = 5\), \(5^1 \times 7^1 = 35\)
- For \(a = 1\): \(3^1 \times 5^0 \times 7^0 = 3\), \(3^1 \times 5^0 \times 7^1 = 21\), \(3^1 \times 5^1 \times 7^0 = 15\), \(3^1 \times 5^1 \times 7^1 = 105\)
- For \(a = 2\): \(3^2 \times 5^0 \times 7^0 = 9\), \(3^2 \times 5^0 \times 7^1 = 63\), \(3^2 \times 5^1 \times 7^0 = 45\), \(3^2 \times 5^1 \times 7^1 = 315\)
- For \(a = 3\): \(3^3 \times 5^0 \times 7^0 = 27\), \(3^3 \times 5^0 \times 7^1 = 189\), \(3^3 \times 5^1 \times 7^0 = 135\), \(3^3 \times 5^1 \times 7^1 = 945\)
- For \(a = 4\): \(3^4 \times 5^0 \times 7^0 = 81\), \(3^4 \times 5^0 \times 7^1 = 567\), \(3^4 \times 5^1 \times 7^0 = 405\), \(3^4 \times 5^1 \times 7^1 = 2835\)

### Step 6: Filter Factors of the Form \(3m + 2\)

Now we filter the factors to find those of the form \(3m + 2\):
- \(5\) (which is \(3 \times 1 + 2\))
- \(35\) (which is \(3 \times 11 + 2\))

### Step 7: Calculate the Sum

Now we sum the factors that are of the form \(3m + 2\):
\[ 5 + 35 = 40 \]

### Step 8: Express in the Form \(8p + q\)

We express \(40\) in the form \(8p + q\):
\[ 40 = 8 \times 5 + 0 \]
Thus, \(p = 5\) and \(q = 0\).

### Step 9: Calculate \(|p - q|\)

Now we find \(|p - q|\):
\[ |p - q| = |5 - 0| = 5 \]

### Final Answer

The value of \(|p - q|\) is:
\[ \boxed{5} \]