Let `S= {1,2,3,4 5,6, 7}`. Then the number of possible functions `f: S rarr S` such that f(m.n)= f(m).f(n) for every `m, n in S and m.n in S` is equal to ____

Answer

To solve the problem, we need to find the number of functions \( f: S \to S \) such that \( f(m \cdot n) = f(m) \cdot f(n) \) for all \( m, n \in S \) where \( m \cdot n \in S \). Here, \( S = \{1, 2, 3, 4, 5, 6, 7\} \).

### Step-by-Step Solution:

1. **Identify the Function Property**:
The property \( f(m \cdot n) = f(m) \cdot f(n) \) indicates that \( f \) is a multiplicative function. This means that the function's behavior is determined by its values at the elements of \( S \).

2. **Evaluate \( f(1) \)**:
Let's first consider \( m = 1 \):
\[ f(1 \cdot n) = f(1) \cdot f(n) \]
This simplifies to:
\[ f(n) = f(1) \cdot f(n) \]
Rearranging gives:
\[ f(n) (1 - f(1)) = 0 \]
Therefore, either \( f(n) = 0 \) (which is not possible since \( f: S \to S \)) or \( f(1) = 1 \).

3. **Evaluate \( f(2) \)**:
Now, let’s set \( m = n = 2 \):
\[ f(2 \cdot 2) = f(2) \cdot f(2) \]
This means:
\[ f(4) = f(2)^2 \]
Since \( f(4) \) must be in \( S \), \( f(2) \) can only take values such that \( f(2)^2 \) is also in \( S \).

4. **Possible Values for \( f(2) \)**:
The possible values for \( f(2) \) that keep \( f(4) \) within \( S \) are:
- If \( f(2) = 1 \), then \( f(4) = 1 \).
- If \( f(2) = 2 \), then \( f(4) = 4 \).
- If \( f(2) = 3 \), then \( f(4) = 9 \) (not allowed since 9 is not in \( S \)).
Thus, \( f(2) \) can be either 1 or 2.

5. **Evaluate \( f(3) \)**:
Next, consider \( m = 2 \) and \( n = 3 \):
\[ f(2 \cdot 3) = f(2) \cdot f(3) \]
This gives:
\[ f(6) = f(2) \cdot f(3) \]
The possible values for \( f(3) \) depend on the value of \( f(2) \).

6. **Possible Values for \( f(3) \)**:
- If \( f(2) = 1 \), \( f(6) = f(3) \) can be any value in \( S \) (1 to 7).
- If \( f(2) = 2 \), then \( f(6) = 2 \cdot f(3) \) must also be in \( S \). Thus, \( f(3) \) can be 1, 2, or 3 (since \( 2 \cdot 4 = 8 \) is not allowed).

7. **Evaluate \( f(5) \) and \( f(7) \)**:
Both \( f(5) \) and \( f(7) \) can take any value from 1 to 7, independent of the previous values.

8. **Count the Total Functions**:
- If \( f(2) = 1 \): \( f(3), f(5), f(6), f(7) \) can each be any of 1 to 7. Therefore, \( 7^4 = 2401 \) functions.
- If \( f(2) = 2 \): \( f(3) \) can be 1, 2, or 3 (3 choices), and \( f(5) \) and \( f(7) \) can each be any of 1 to 7. Therefore, \( 3 \cdot 7^2 \cdot 7 = 147 \) functions.

9. **Total Functions**:
Adding both cases gives:
\[ 2401 + 147 = 2548 \]

Thus, the total number of possible functions \( f: S \to S \) satisfying the given conditions is **2548**.