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Yara Leonard
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Determine the order of every element of Z24
Group elements of Z24 refer to the set of integers from 0 to 23 that form a group under the operation of addition modulo 24. This means that when two elements are added together, the result will always be within the set of integers from 0 to 23.
There are 24 group elements in Z24, as the set includes all integers from 0 to 23.
The order of a group element in Z24 is the number of times the element must be added to itself to get the identity element, which is 0. For example, the order of 3 in Z24 is 8, as 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 24, which is equivalent to 0 in Z24.
To find the order of a group element in Z24, you can use the formula n/gcd(n, 24), where n is the smallest positive integer such that n * element = 0, and gcd(n, 24) is the greatest common divisor of n and 24. For example, to find the order of 3 in Z24, we can use the formula 24/gcd(24, 3) = 24/3 = 8.
No, the order of a group element in Z24 cannot be greater than 24. This is because the set only includes integers from 0 to 23, and any element added to itself more than 24 times will result in a number greater than 23, which is not within the set of group elements in Z24.