Question

Let $X$ be the set of all positive integers greater than or equal to 8 and let $f: X \rightarrow X$ be a function such that $f(x+y) = f(xy)$ for all $x \ge 4, y \ge 4$, if $f(8)=9$. Then number of positive integer factors of $f(9)$ will be

Answer 1

3

Answer 2

The given condition is $f(x+y) = f(xy)$ for all $x \ge 4, y \ge 4$.

1. Using $x=4, y=4$: $f(4+4) = f(4 \times 4) \implies f(8) = f(16)$. Since $f(8)=9$, we have $f(16)=9$.
2. Using $x=4, y=5$: $f(4+5) = f(4 \times 5) \implies f(9) = f(20)$.
3. We use $f(x+y) = f(xy)$ with $x=4, y=16$ (since $4 \ge 4, 16 \ge 4$): $f(4+16) = f(4 \times 16) \implies f(20) = f(64)$.
4. We know $f(16)=9$. We can write $16 = 8+8$ where $8 \ge 4$. So, $f(8+8) = f(8 \times 8) \implies f(16) = f(64)$. Thus, $f(64)=9$.
5. From step 3, $f(20) = f(64) = 9$.
6. From step 2, $f(9) = f(20) = 9$.
7. The number of positive integer factors of $f(9)=9$ are 1, 3, and 9. There are 3 factors.