I needed help in classifying the following quotient groups according to the fundamental theorem of finitely generated abelian groups:
$$ \begin{array} &(\mathbb Z_4 \times \mathbb Z_{16})/\langle(1, 4)\rangle,\\ (\mathbb Z_4 \times\mathbb Z_{16})/\langle(2, 4)\rangle,\\ (\mathbb Z \times \mathbb Z \times \mathbb Z)/\langle(1, 2, 4)\rangle. \end{array} $$
What I tried out:
(i) $F : \mathbb Z_4 \times \mathbb Z_{16}\longrightarrow \mathbb Z_{16}$ defined by $F(a, b) = 4a - b \mod 16$ is a well-defined surjective homomorphism with $\ker F = \langle(1, 4)\rangle.$
$F$ is well-defined: Writing $a + 4j,$ and $b + 16k$ for any integers $j, k,$ we have $$4(a + 4j) - (b + 16k) = (4a - b) + 16(j-k) = 4a - b \mod 16$$ for any $j, k.$
$F$ is a homomorphism: For any $(a, b), (c, d)$ in $\mathbb Z_4 \times \mathbb Z_{16},$ we have
$$F(a, b) + F(c, d) = (4a - b) + (4c - d) = 4(a + c) - (b + d) = F(a+c, b+d).$$
$F$ is surjective: For any $c$ in $\mathbb Z_{16},$ we have $$F(0, -c) = 4 \cdot 0 - (-c) = c,$$ as required.