(What is the value of 23 mod 6?)

The expression "51 mod 11" asks for the remainder when 51 is divided by 11. This is a standard modular arithmetic operation. To calculate this, we find the largest multiple of 11 that is less than or equal to 51. In this case, 11 multiplied by 4 is 44. We then subtract this multiple from 51 to find the remainder: 51 - 44 = 7. Therefore, the value of 51 mod 11 is 7.

A. The remainder is not 4. 11 4 + 4 = 48, not 51.

B. The remainder is not 5. 11 4 + 5 = 49, not 51.

D. The result of a modulo n operation must be an integer in the range [0, n-1]. 11 is outside the range [0, 10].

1. Rosen, K. H. (2019). Discrete Mathematics and Its Applications (8th ed.). McGraw-Hill. Chapter 4, Section 4.1, "Divisibility and Modular Arithmetic," Definition 4, p. 271.

2. Paar, C., & Pelzl, J. (2010). Understanding Cryptography: A Textbook for Students and Practitioners. Springer. Chapter 4, Section 4.2, "The Integers," pp. 88-89.

3. MIT OpenCourseWare. (2015). Mathematics for Computer Science (6.042J). Fall 2010. Lecture 10: Number Theory I, Section 2.1 "The Modulo Operator."