Q2
A function f maps the set of natural numbers to whole numbers, such that f(xy) = f(x)f(y) + f(x) + f(y) for all x, y and f(p) = 1 for every prime number p. Then, the value of f(160000) is
Answer: C
Asked: \(f(160000)\) where \(f(xy)=f(x)f(y)+f(x)+f(y)\) and \(f(p)=1\) for every prime p.
Key idea: add 1 to both sides: \(f(xy)+1=(f(x)+1)(f(y)+1)\). So \(g(x)=f(x)+1\) is multiplicative: \(g(xy)=g(x)g(y)\), with \(g(p)=2\) for every prime.
Hence \(g(n)=2^{k}\), where \(k\) is the number of prime factors of \(n\) counted with repetition.
\(160000=16\times10^4=2^4\times2^4\times5^4=2^8\times5^4\), so \(k=8+4=12\).
\(g(160000)=2^{12}=4096\Rightarrow f(160000)=4095\).
Answer: 4095