Understanding the Remainder of 11^77^11 when Divided by 18

Modular arithmetic is a fundamental concept in number theory and is essential for solving a wide range of mathematical and computational problems. One common application is finding remainders when numbers are divided by a specific modulus. In this article, we will explore the process of determining the remainder when (11^{77^{11}}) is divided by 18. This involves understanding and utilizing properties of modular arithmetic and simplifying the expression through various congruence relations.

Introduction to Modular Arithmetic

Modular arithmetic, or modulo, is a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value—the modulus. For example, in modulo 18, we are interested in the remainders when numbers are divided by 18. The notation (a equiv b pmod{m}) means that (a) and (b) have the same remainder when divided by (m).

Simplifying the Expression

We start with the given expression, 117711. To simplify this, we will use properties of exponents and modulus to reduce the base and the exponent. Let's break down the problem step by step.

Step 1: Simplify the Exponent

First, we note that the exponent (77^{11}) is a large number, making direct computation infeasible. We need to simplify it modulo 18. We begin by using the fact that:

[ 7^3 equiv 343 equiv 1 pmod{18} ]

This tells us that every third power of 7 is congruent to 1 modulo 18. Hence, we can write:

[ 77 equiv 77 - 3 cdot 25 equiv 7 pmod{3} ]

Since (77 3 cdot 25 2), we have:

[ 77 equiv 2 pmod{3} ]

This simplifies our exponent to a more manageable form. We now need to find:

[ 77^{11} equiv 2^{11} pmod{18} ]

Step 2: Simplify the Base

Next, we note that:

[ 11 equiv -7 pmod{18} ]

Therefore, we have:

[ 11^{77^{11}} equiv (-7)^{77^{11}} pmod{18} ]

Step 3: Use Congruence Relations

Now we know:

[ (-7)^{3 times 21} equiv (-1)^2 equiv 1 pmod{18} ]

Since (77 3 times 21 4), we have:

[ 77^{11} equiv 2^{11} equiv 2^{11 - 3 times 7} cdot 2^4 equiv 2^4 equiv 16 pmod{18} ]

Thus:

[ 11^{77^{11}} equiv (-7)^{16} pmod{18} ]

Step 4: Simplify Further

We know:

[ 7^3 equiv 1 pmod{18} ]

This means that every third power of 7 is congruent to 1, and hence:

[ 7^{15} equiv 1 pmod{18} ]

Therefore:

[ 7^{16} equiv 7 cdot 7^{15} equiv 7 cdot 1 equiv 7 pmod{18} ]

Thus:

[ (-7)^{16} equiv (-7) cdot 7^{15} equiv (-7) cdot 1 equiv -7 equiv 11 pmod{18} ]

Conclusion

In conclusion, the remainder when (11^{77^{11}}) is divided by 18 is 6. This solution demonstrates the power of modular arithmetic and the importance of simplifying expressions using congruence relations and properties of exponents.

Understanding modular arithmetic and its applications is crucial in various fields, including cryptography, computer science, and number theory. This problem serves as an excellent example of how to break down complex expressions and apply modular arithmetic to find elegant solutions.

Keywords: remainder, modular arithmetic, division by 18