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 $x \geq 4, y \geq 4$, if $f(8) = 9$. Then number of positive integer factors of $f(9)$ will be.

Answer 1

3

Answer 2

The functional equation is $f(x+y) = f(xy)$ for $x \geq 4, y \geq 4$. We are given $f(8) = 9$. Using the functional equation with $x=4, y=5$, we get $f(4+5) = f(4 \times 5)$, so $f(9) = f(20)$. Using $x=4, y=16$, we get $f(4+16) = f(4 \times 16)$, so $f(20) = f(64)$. Using $x=8, y=8$, we get $f(8+8) = f(8 \times 8)$, so $f(16) = f(64)$. Using $x=4, y=4$, we get $f(4+4) = f(4 \times 4)$, so $f(8) = f(16)$. Combining these equalities, we have $f(9) = f(20) = f(64) = f(16) = f(8)$. Since $f(8) = 9$, it follows that $f(9) = 9$. The number of positive integer factors of 9 (which is $3^2$) is $(2+1) = 3$. The factors are 1, 3, and 9.