Find all prime ideals and all maximal ideals of Z12

Note that the ideals of Z₁₂ are Z₁₂, 2Z₁₂, 3Z₁₂, 4Z₁₂, 6Z₁₂, and 12Z₁₂ = {0}

Looking at the quotients:
Z₁₂ / Z₁₂ ≅ {0}
Z₁₂ / 2Z₁₂ ≅ Z₆
Z₁₂ / 3Z₁₂ ≅ Z₄
Z₁₂ / 4Z₁₂ ≅ Z₃, integral domain/field
Z₁₂ / 6Z₁₂ ≅ Z₂, integral domain/field
Z₁₂ / {0} ≅ Z₁₂

Hence, only 4Z₁₂ and 6Z₁₂ are prime/maximal.

I hope this helps!