1. **State the problem:** We have a map $\rho: \mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ defined by $\rho(a + bi) = [a + 7b]$. We want to find the kernel of this homomorphism. 2. **Recall the kernel definition:** The kernel of a homomorphism $\rho$ is the set of all elements in the domain that map to the zero element in the codomain. Here, zero in $\mathbb{Z}/10\mathbb{Z}$ is $[0]$. 3. **Set up the kernel condition:** $$\ker(\rho) = \{a + bi \in \mathbb{Z}[i] : \rho(a + bi) = [0]\}$$ 4. **Apply the definition of $\rho$:** $$\rho(a + bi) = [a + 7b] = [0] \implies a + 7b \equiv 0 \pmod{10}$$ 5. **Interpret the congruence:** This means $a + 7b$ is divisible by 10. So the kernel is all Gaussian integers $a + bi$ such that $a + 7b$ is a multiple of 10. 6. **Express kernel explicitly:** $$\ker(\rho) = \{a + bi \in \mathbb{Z}[i] : a + 7b = 10k, k \in \mathbb{Z}\}$$ 7. **Summary:** The kernel consists of all Gaussian integers whose real part plus seven times the imaginary part is divisible by 10. **Final answer:** $$\ker(\rho) = \{a + bi \in \mathbb{Z}[i] : a + 7b \equiv 0 \pmod{10}\}$$