While the "solution" below is correct, the reasoning is getting there: we actually need to check the definition, which requires that c(mn)=c(m)*c(n) when gcd(m,n)=1. This person has correctly observed that when we compute sigma(m) and sigma(n), there's no guarantee that those will be relatively prime, which may make multiplicativity (oooh!) fail.
In fact, a counter-example is the easiest way to go: it turns out (for example), that
By the way, you might start by looking for the counter-example using the smallest values possible, 2, 3, 6: "multiplicativity" works for that choice!;)
