StatusX said:Note that phi(1) determines phi for all integers. There is a simple relation between the order of phi(1) and the kernel of phi. By the way, you don't have enough information to get one answer, but you can narrow down the possibilities.
rocky926 said:I don't know, I am totally confused right now... I would guess the identity element is 1.
Dick said:I was assuming the statement of the question was not only stating the domain and range but that literally phi(z)=(z mod 24)x(z mod 81). Do you think I'm wrong? I already gaffed one.
StatusX said:Maybe you're right. Usually the notation f:A->B just specifies the domain and range though. If you're right, then the notation the OP's using is so economical as to be confusing.
The Kernel of Z (mod 24) X Z (mod 81) is a mathematical concept that refers to the set of all pairs (a,b) where a is an integer between 0 and 23 and b is an integer between 0 and 80. This set represents the elements that map to the identity element (0,0) in the group Z (mod 24) X Z (mod 81).
The Kernel of Z (mod 24) X Z (mod 81) is calculated by finding all pairs (a,b) that satisfy the equation a mod 24 = 0 and b mod 81 = 0. This can be done by listing out all possible pairs and checking if they satisfy the equation, or by using modular arithmetic and finding the values of a and b that make the equation true.
The cardinality (or size) of the Kernel of Z (mod 24) X Z (mod 81) is equal to the number of elements in the set. In this case, the cardinality is equal to the product of the cardinalities of Z (mod 24) and Z (mod 81). Therefore, the cardinality is equal to 24 x 81 = 1944.
The Chinese Remainder Theorem states that for any set of pairwise coprime integers, there exists a unique solution to a system of congruences. In the case of Z (mod 24) X Z (mod 81), the integers 24 and 81 are not coprime, but their greatest common divisor is 3. Therefore, the Kernel of Z (mod 24) X Z (mod 81) can be seen as the set of all solutions to the system of congruences x mod 24 = 0 and x mod 81 = 0, which is a special case of the Chinese Remainder Theorem.
The Kernel of Z (mod 24) X Z (mod 81) is significant in modular arithmetic because it represents the elements that map to the identity element in the group Z (mod 24) X Z (mod 81). This set can also be thought of as the "null space" or "zero set" of this group, and understanding its properties can help in solving equations and systems of congruences within this group.