How many subgroups of z contain 12z?

Answer 1

The group of integers ( \mathbb{Z} ) contains the subgroup ( 12\mathbb{Z} ), which consists of all multiples of 12. The subgroups of ( \mathbb{Z} ) are of the form ( n\mathbb{Z} ) for some integer ( n ). The subgroups of ( 12\mathbb{Z} ) correspond to the divisors of 12, which are 1, 2, 3, 4, 6, and 12, resulting in 6 distinct subgroups. Therefore, there are 6 distinct subgroups of ( \mathbb{Z} ) that contain ( 12\mathbb{Z} ).