Characteristic of ring $R_1\otimes\ldots\otimes R_k$

Question

Let $R_1,\ldots, R_k$ be unital rings and $\otimes=\otimes_\mathbb{Z}$ and $\mathrm{chr}$ the characteristic. How can one see that $$\mathrm{chr}(R_1\otimes\ldots\otimes R_k)=\gcd(\mathrm{chr}R_1,\ldots,\mathrm{chr}R_k)?$$

Edit: additionally assume that $\mathrm{chr}R_1,\ldots,\mathrm{chr}R_k$ are all nonzero.