Let fourth therm of an arithmetic progression be 6 and `m^(th)` term be 18. If A.P has intergal terms only then the numbers of such A.P s is `"____________"`

Explanation

To solve the problem, we need to find the number of arithmetic progressions (APs) that satisfy the given conditions. Let's break down the solution step by step.

### Step 1: Understand the given information
We know that:
- The fourth term of the AP (denoted as \( A_4 \)) is 6.
- The \( m^{th} \) term of the AP (denoted as \( A_m \)) is 18.

### Step 2: Write the formulas for the terms
The \( n^{th} \) term of an arithmetic progression can be expressed as:
\[ A_n = A + (n-1)D \]
where \( A \) is the first term and \( D \) is the common difference.

### Step 3: Set up the equations
From the information given:
1. For the fourth term:
\[ A_4 = A + 3D = 6 \quad \text{(1)} \]
2. For the \( m^{th} \) term:
\[ A_m = A + (m-1)D = 18 \quad \text{(2)} \]

### Step 4: Express \( A \) in terms of \( D \)
From equation (1), we can express \( A \) in terms of \( D \):
\[ A = 6 - 3D \quad \text{(3)} \]

### Step 5: Substitute \( A \) in equation (2)
Substituting equation (3) into equation (2):
\[ (6 - 3D) + (m-1)D = 18 \]
Simplifying this gives:
\[ 6 - 3D + mD - D = 18 \]
\[ 6 + (m - 4)D = 18 \]
\[ (m - 4)D = 12 \quad \text{(4)} \]

### Step 6: Analyze equation (4)
From equation (4), we can see that \( m - 4 \) must divide 12. We will find the divisors of 12 to determine possible values for \( m - 4 \).

### Step 7: Find the divisors of 12
The divisors of 12 are:
\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \]
This gives us the possible values for \( m - 4 \):
- \( m - 4 = 1 \) → \( m = 5 \)
- \( m - 4 = -1 \) → \( m = 3 \)
- \( m - 4 = 2 \) → \( m = 6 \)
- \( m - 4 = -2 \) → \( m = 2 \)
- \( m - 4 = 3 \) → \( m = 7 \)
- \( m - 4 = -3 \) → \( m = 1 \)
- \( m - 4 = 4 \) → \( m = 8 \)
- \( m - 4 = -4 \) → \( m = 0 \) (not valid since \( m \) must be positive)
- \( m - 4 = 6 \) → \( m = 10 \)
- \( m - 4 = -6 \) → \( m = -2 \) (not valid)
- \( m - 4 = 12 \) → \( m = 16 \)
- \( m - 4 = -12 \) → \( m = -8 \) (not valid)

### Step 8: Valid values for \( m \)
The valid values for \( m \) are:
- 5, 3, 6, 2, 7, 1, 8, 10, 16

### Step 9: Count the valid values
Counting these valid values gives us a total of 9 valid values for \( m \).

### Final Answer
Thus, the number of such arithmetic progressions is:
\[ \boxed{9} \]