8. Caesar's encryption method can be represented by the function, \( f \), that assigns to the nonnegative integer \( p, p \leq 25 \), the integer \( f(p) \) in the set \( \{0,1,2, \ldots, 25\} \) with \[ f(p)=(p-5) \bmod 26 \text {. } \] Encode the phrase, "I WANT AN A".

To encode the phrase "I WANT AN A" using Caesar's encryption method defined by the function \( f(p) = (p - 5) \mod 26 \), we first need to convert each letter into its corresponding nonnegative integer value, where \( A = 0, B = 1, C = 2, \ldots, Z = 25 \). Here are the integer values for the letters in the phrase "I WANT AN A": - I = 8 - W = 22 - A = 0 - N = 13 - T = 19 - (space) = not encoded - A = 0 - N = 13 - (space) = not encoded - A = 0 Now we will apply the function \( f(p) = (p - 5) \mod 26 \) to each letter: 1. For I (8): \[ f(8) = (8 - 5) \mod 26 = 3 \] (which corresponds to D) 2. For W (22): \[ f(22) = (22 - 5) \mod 26 = 17 \] (which corresponds to R) 3. For A (0): \[ f(0) = (0 - 5) \mod 26 = 21 \] (which corresponds to V) 4. For N (13): \[ f(13) = (13 - 5) \mod 26 = 8 \] (which corresponds to I) 5. For T (19): \[ f(19) = (19 - 5) \mod 26 = 14 \] (which corresponds to O) 6. For A (0): \[ f(0) = (0 - 5) \mod 26 = 21 \] (which corresponds to V) 7. For N (13): \[ f(13) = (13 - 5) \mod 26 = 8 \] (which corresponds to I) 8. For A (0): \[ f(0) = (0 - 5) \mod 26 = 21 \] (which corresponds to V) Now, we can put it all together, replacing each letter with its encoded counterpart: - I → D - W → R - A → V - N → I - T → O - (space) → (space) - A → V - N → I - (space) → (space) - A → V Thus, the encoded phrase "I WANT AN A" becomes: **"D R V I O V I V"**