I want to solve this question, but I have no idea where should I start. I'm not sure even if I understand the question correctly or not. The question is:
Let $X$ be a path connected and locally path connected space, and let $Y=S^1 \times S^1 \times \ldots \times S^1$, a product of $n$ copies of the circle with $n\ge 1$. Show that if the fundamental group of $X$ at $x_0$ is finite, then every map $f:X\to Y$ is null-homotopic.