Question

The following program uses function multiple to determine whether the integer entered from the keyboard is a multiple of some integer \(x\). Examine function multiple, then determine the value of \(x\). 1 // Exercise 22.19: ex22_19.cpp 2 // This program determines if a value is a multiple of X. 3 #include 4 5 using std::cout; 6 using std::cin; 7 using std::endl; 8 9 bool multiple( int ); 10 11 int main() 12 { 13 int y; 14 15 cout << "Enter an integer between 1 and 32000: "; 16 cin >> y; 17 18 if ( multiple( y ) ) 19 cout << y << " is a multiple of X" << endl; 20 else 21 cout << y << " is not a multiple of X" << endl; 22 23 return 0; 24 25 } // end main 26 27 // determine if num is a multiple of X 28 bool multiple( int num ) 29 { 30 bool mult = true; 31 32 for ( int i = 0, mask = 1; i < 10; i++, mask <<= 1 ) 33 34 if ( ( num & mask ) != 0 ) { 35 mult = false; 36 break; 37 38 } // end if 39 40 return mult; 41 42 } // end function multiple

Short answer

The integer `x` is 1024.

Step-by-step solution

Understanding the multiple Function

The function `multiple` checks if the input integer, `num`, is a multiple of some integer `x`. The variable `mask` is initialized to 1 and left-shifted in each iteration of the loop.

Loop Execution for Bitwise Operations

The loop iterates 10 times with `i` starting at 0 and ending at 9, while multiplying `mask` by 2 (left shift) in each loop iteration. This means `mask` takes on values 1, 2, 4, 8, ..., 512 (powers of 2 up to 2^9). In binary, these values represent the checks for bits at each position from 0 to 9.

Analyzing Bitwise AND Operation

The expression `(num & mask) != 0` checks if any of the first 10 bits (from least significant bit) of `num` are set. If `num` has any of these bits set, it is not a multiple of `x`, and `mult` is set to false.

Determining the Value of x

For `mult` to remain true (indicating `num` is a multiple of `x`), none of the bits accessed by `mask` should be set. This implies `num` must be 0 for mask = 1, 2, ..., 512, meaning `num` must be a multiple of 1024 (since `mask <<= 1` proceeds through 2^0 to 2^9).

Conclusion

Therefore, `x` is 1024 because only a multiple of 1024 will have clear (zero) bits in all positions 0 to 9, successfully resulting in `mult` remaining true.

Key concepts

Bitwise AND

The bitwise AND operation is a fundamental concept in programming that allows you to perform bit-level operations on integers. When you apply the bitwise AND operator between two binary numbers, it compares each bit of the two numbers. If both bits at the same position are 1, the result will be 1; otherwise, it will be 0.
For example:

• Consider the numbers 5 (0101 in binary) and 3 (0011 in binary).
• The bitwise AND of 5 and 3 is 1 (0001 in binary).

In the provided C++ code, the expression `(num & mask)` is critical. Here, `num` is the number entered by the user, and `mask` is updated in a loop to test individual bits of `num`. If any of the bits set by `mask` in `num` results in a non-zero value, `(num & mask) != 0` evaluates to true. Thus, `mult` is set to `false`, indicating that `num` is not a multiple of `x`. This operation is efficient for checking specific bit positions in binary numbers. Understanding bitwise AND is essential for optimizing programs and manipulating binary data efficiently.

Left Shift Operator

The left shift operator (<<) is used in C++ to shift the bits of a number to the left by a specified number of positions, effectively multiplying the number by 2 for each shift. This operation is equivalent to multiplying the original number by powers of 2.
For instance:

• Shifting the binary number `0001` (which is 1 in decimal) by one position to the left results in `0010`, which is 2.
• Shifting `0010` by one more position yields `0100`, which is 4.

In the provided solution, `mask` is initially set to 1 and is left-shifted once per iteration in the loop using `mask <<= 1`. This makes `mask` take on values such as 1, 2, 4, 8, and so on, up to 512. It effectively multiplies the mask by 2 in each loop iteration. This escalating value allows the program to check each of the first 10 bits of `num`, helping to determine if they are set to 1 or 0. By understanding the left shift operator, you can harness its power to efficiently perform multiplication and bitwise analysis in programming tasks.

Control Structures

Control structures in C++ define the flow of control in a program, allowing you to dictate how and when particular blocks of code execute. The most common control structures are loops and conditional statements. In the given exercise, a `for` loop and an `if` statement are used.

• The `for` loop allows repeated execution of a block of code. Here, it iterates 10 times to test each bit of `num` using `mask`.
• `for (int i = 0, mask = 1; i < 10; i++, mask <<= 1)` initializes the loop with `i` at 0 and `mask` at 1, continuing until `i` is less than 10.

The `if` statement evaluates whether `(num & mask) != 0`, checking if any of the bits tested by the current `mask` are set in `num`. If this condition is true, the code inside the `if` block is executed, setting `mult` to `false` and breaking the loop prematurely. Understanding control structures is crucial because they determine how functions respond to different inputs, effectively managing program logic and flow. This exercise demonstrates their utility in bitwise manipulation scenarios.

Function Programming

Function programming in C++ involves writing reusable code modules to perform specific tasks. Functions help organize code into manageable pieces, making it easier to read, maintain, and debug.
In the provided exercise, the function `multiple` plays a vital role. It checks whether a number is a multiple of a specific integer `x`.

• The function is defined with `bool multiple(int num)`, indicating that it returns a boolean value based on the test results.
• In the `main` function, `multiple(y)` is called and determines if the input integer `y` is a multiple of 1024 as `x` is determined to be 1024.

This example demonstrates how function programming encapsulates logic and simplifies solutions by breaking down tasks into smaller, specific problems that a function can easily address. This modular approach promotes code reusability and minimizes redundancy, improving program structure and efficiency.