Question

For a positive integer $x$, define $f(x)$ such that $f(x+a) = f(ax)$, where $a$ is an integer and $f(1)=4$. If $f(1003)=k$, then the value of $k$ will be:

• A. 1003
• B. 1004
• C. 1005
• D. 1006
• E. None of the above

Hint:

When given a functional equation, test with small substitutions (like $x=1$) to reduce the form. If all values become equal, the function must be constant.

Answer 1

Answer: (E)

Step 1: Analyze the functional equation.
We are given: \[ f(x+a) = f(ax). \] This means we can {link addition with multiplication}. Step 2: Try with small values.
Take $x=1$ and $a=n-1$. Then \[ f(1+(n-1)) = f((n-1)\cdot 1). \] So, \[ f(n) = f(n-1). \] But we must be careful—let’s apply systematically. Step 3: Express $f(n)$.
From the equation: \[ f(x+a) = f(ax). \] Take $x=1$: \[ f(1+a) = f(a). \] This shows $f(n+1)=f(n)$ for all $n$. Hence $f$ is {constant}. Step 4: Value of the constant.
Since $f(1)=4$, it follows that: \[ f(n)=4 \quad \forall n. \] Step 5: Apply to the problem.
So, \[ f(1003)=4 \quad \Rightarrow \quad k=4. \] Thus the answer is not among 1003, 1004, 1005, or 1006. \[ \boxed{k=4} \]