The number of group homomorphisms from ℤ/150ℤ to ℤ/90ℤ is 

Concept:

If  and  are cyclic groups, the number of group homomorphisms from  to  is given by

 where  is the greatest common divisor of  n and m.

Explanation:

The number of group homomorphisms from  to  is given by the greatest common divisor (gcd)

of the orders of the two groups. That is  
  
Prime factorization of 150:

Prime factorization of 90:

Now, the gcd of 150 and 90 is the product of the lowest powers of the common prime factors:
   


The number of group homomorphisms from  to  is 30.

Thus, option 1) is correct.