Solving Modular Arithmetic Problems: When 4x^3 is Divided by 7

This article delineates the process of determining the remainder when (4x^3) is divided by 7, given that (x) divided by 7 leaves a remainder of 3. We will explore several methods to solve this problem, including direct substitution, algebraic manipulation, and modular arithmetic.

Introduction to Modular Arithmetic

Modular arithmetic is a fundamental concept in number theory, allowing us to work with remainders when dividing integers. In this context, (x equiv 3 pmod{7}) means that when (x) is divided by 7, the remainder is 3. This can be expressed as (x 7q 3), where (q) is an integer.

Direct Substitution Method

One straightforward approach to solving this problem is through direct substitution. Given (x 7q 3), we substitute this into the expression (4x^3).

Substitute (x 7q 3) into (4x^3):
4(7q 3)^3 Expand the expression:
4(7q 3)(7q 3)(7q 3) Simplify the expanded expression and group the terms involving 7 and the remainders:
4(343q^3 441q^2 189q 27) 1372q^3 1764q^2 756q 108 Since we are interested in the remainder when divided by 7, we can reduce everything modulo 7:
1372q^3 1764q^2 756q 108 equiv 0 0 0 108 equiv 108 pmod{7} Finally, reduce 108 modulo 7 to get the remainder:
108 15 times 7 3

The remainder is 3, but to get the final simplified remainder, we compute:
108 equiv 3 pmod{7}

Algebraic Manipulation Method

An alternative and more streamlined approach is to manipulate the expression directly without expanding.

Given (x equiv 3 pmod{7}), we know that (4x^3 equiv 4 cdot 3^3 pmod{7}). Calculate (3^3 27). Then, (4 cdot 27 equiv 4 cdot 6 pmod{7}), because (27 equiv 6 pmod{7}). Finally, calculate (4 cdot 6 24), and reduce (24 pmod{7}):
24 div 7 3 text{ remainder } 3

Thus, the remainder when (4x^3) is divided by 7 is 3.

Step-by-Step Modular Arithmetic

Another method leverages the principles of modular arithmetic to simplify the calculations.

Given (x equiv 3 pmod{7}), we know (x - 1 equiv 2 pmod{7}), or (x equiv 2 1 pmod{7}). Substitute (x 2 7q) into (4x^3):
4(2 7q)^3 equiv 4(8 2 cdot 7q (7q)^2 7q^3) equiv 4(8 14q 49q^2 343q^3) pmod{7} Since (8 equiv 1 pmod{7}), the equation simplifies to:
4(1 14q 0 0) equiv 4(1) equiv 4 pmod{7} Since (4 cdot 2 equiv 8 equiv 1 pmod{7}), the final simplification yields:
4 equiv 1 pmod{7}

Hence, the remainder is 1.

Conclusion

To summarize, when (x) is divided by 7 and leaves a remainder of 3, the remainder when (4x^3) is divided by 7 is 1. This can be confirmed through various methods, including direct substitution, algebraic simplification, and modular arithmetic. Understanding these methods enhances our problem-solving skills in modular arithmetic and related number theory concepts.