Consider the mathematical puzzle, 'If 42x5 is a multiple of 9 and x is a digit, then what is the value of x?' This problem requires an understanding of the rules governing multiples of 9 and basic algebraic manipulation. Below, the solution to this problem is explained step-by-step, with an exploration of the underlying principles.

Introduction

A number is considered a multiple of 9 if the sum of its digits is a multiple of 9. For example, if 42x5 is to be a multiple of 9, then the sum of its digits (4, 2, x, 5) must also be a multiple of 9.

Step-by-Step Solution

The initial approach involves setting up the equation based on the rule that the sum of the digits must be a multiple of 9. Given that 42x5 must be a multiple of 9, the sum of the digits is 4 2 x 5, which simplifies to 11 x.

Method 1: Direct Sum Check

The simplest method involves direct calculation:
4 2 x 5 11 x
For 11 x to be a multiple of 9, x must be 7 because 11 7 18, a multiple of 9.
Therefore, x 7.

Method 2: Algebraic Verification

Alternatively, one can set up an equation where the sum of the digits equals a multiple of 9:
4 2 x 5 11 x 9t (where t is an integer)
The value of x can be found by setting the equation to 9 and solving for x:
11 x 9
x 7.
Moreover, substituting x back into the original number 4275, we can verify:

4 2 7 5 18, which is a multiple of 9, confirming the solution.

Common Misinterpretations

There are also approaches that attempt to solve the equation differently, such as introducing unnecessary operations or misinterpreting the problem. For instance:

5x 9 42
5x 42 - 9
x 33/5 (which would result in a non-integer value for x, contradicting the condition that x is a digit).

Starting from the equation 5x 9 42, subtract 9 from both sides to get 5x 33, then divide by 5 to get
x 33/5
This will yield a non-integer, which is not suitable for the problem constraints.

Conclusion: The correct value of x that makes 42x5 a multiple of 9 is 7. This is confirmed through multiple valid algebraic methods and through the verification of the sum of the digits.

Further Practice and Exploration

This problem is an excellent example for students and professionals to practice their skills in algebraic manipulation and number theory. It emphasizes the importance of adhering to the governing principles of mathematics, such as the divisibility rule for 9, and demonstrates the application of basic algebra in solving complex problems.