Elements of the group Z50

The group Z50 consists of integers from 0 to 49, with the operation of addition modulo 50. In other words, if we add two integers x and y in Z50, the result is (x + y) mod 50.

Order of an element

The order of an element in a group is the smallest positive integer n such that the element raised to the power of n equals the identity element of the group. In the case of Z50, the identity element is 0.

Finding elements of order 10

To find elements of order 10 in Z50, we need to find the integers x such that x^10 mod 50 = 0.

Step 1: Calculating x^10 mod 50 for different values of x

We can calculate x^10 mod 50 for different values of x to find elements of order 10. Let's calculate x^10 mod 50 for x = 1, 2, 3, ..., 49.

x=1: 1^10 mod 50 = 1 mod 50 = 1
x=2: 2^10 mod 50 = 1024 mod 50 = 24
x=3: 3^10 mod 50 = 59049 mod 50 = 49
x=4: 4^10 mod 50 = 1048576 mod 50 = 26
x=5: 5^10 mod 50 = 9765625 mod 50 = 25
...
x=49: 49^10 mod 50 = 576650390625 mod 50 = 25

Step 2: Identifying elements with order 10

From the calculations above, we can see that the values of x for which x^10 mod 50 = 0 are x = 10, 20, 30, 40.

Step 3: Counting the elements

Since we have found 4 elements (10, 20, 30, 40) that have order 10 in Z50, the answer is (B) 4.

Summary:

- The group Z50 consists of integers from 0 to 49 with addition modulo 50.
- The order of an element in a group is the smallest positive integer n such that the element raised to the power of n equals the identity element of the group.
- To find elements of order 10 in Z50, we calculate x^10 mod 50 for different values of x.
- From the calculations, we find that the elements 10, 20, 30, and 40 have order 10 in Z50.
- Therefore, the answer is (B) 4.