Find [tex]\phi(20000)[/tex] using the formula for the Euler [tex]\phi[/tex]-function. Also, find [tex]\lvert z_{20000} \rvert[/tex] and [tex]\lvert u_{20000} \rvert[/tex].

Options:

A. [tex]\phi(20000) = 8000[/tex], [tex]\lvert z_{20000} \rvert = 20000[/tex], [tex]\lvert u_{20000} \rvert = 19999[/tex]
B. [tex]\phi(20000) = 12000[/tex], [tex]\lvert z_{20000} \rvert = 19999[/tex], [tex]\lvert u_{20000} \rvert = 20000[/tex]
C. [tex]\phi(20000) = 16000[/tex], [tex]\lvert z_{20000} \rvert = 19999[/tex], [tex]\lvert u_{20000} \rvert = 19998[/tex]
D. [tex]\phi(20000) = 20000[/tex], [tex]\lvert z_{20000} \rvert = 19998[/tex], [tex]\lvert u_{20000} \rvert = 19999[/tex]

The Euler phi function gives the count of positive integers less than or equal to n that are relatively prime to n. For ϕ(20000), the value is 8000. The absolute values of z and u are 20000 and 19999, respectively.

Explanation:

The Euler phi function, denoted as ϕ(n), gives the count of positive integers less than or equal to n that are relatively prime to n. In other words, ϕ(n) is the number of positive integers that do not share any common factors with n (except for 1).

Given the formula for the Euler phi function, we can find that ϕ(20000) = 8000.

For part (b), ∣z₂₀₀₀₀∣ refers to the absolute value of z and ∣u₂₀₀₀₀∣ refers to the absolute value of u. The given options state that ∣z₂₀₀₀∣ = 20000 and ∣u₂₀₀₀∣ = 19999.

Therefore, the correct option is a)ϕ(20000)=8000, ∣z₂₀₀₀∣=20000, ∣u₂₀₀₀∣=19999.