Question

Z₅ = {0,1,2,3,4} +, *. In Z₅, 11² * 23² =

A. impossible to tell
B. 4
C. 2
D. 0
E. 64009
F. 3
G. 1
H. -1

Answer 1

11² * 23² in Z₅ is calculated by reducing 11 to 1 and 23 to 3, then squaring and multiplying these results modulo 5. The final answer is 4 in Z₅, making option (B) 4 the correct answer.

Step-by-step explanation:

The student's question involves operations within the finite field Z₅, which is the set {0,1,2,3,4} with modular arithmetic under addition (+) and multiplication (*). In modular arithmetic, numbers wrap around upon reaching the modulus, which is 5 in the case of Z₅. Thus, each operation is performed modulo 5.

To solve 11² * 23² in Z₅, we first find the squares of 11 and 23 modulo 5. Recognize that 11 is congruent to 1 modulo 5 (11 % 5 = 1), and 23 is congruent to 3 modulo 5 (23 % 5 = 3). Squaring these, we get:

• 1² = 1 (mod 5)
• 3² = 9, which is congruent to 4 modulo 5 (9 % 5 = 4)

Multiplying these results together in Z₅:

• 1 * 4 = 4 (mod 5)

Therefore, 11² * 23² = 4 in Z₅, and the correct answer is (B) 4.