Question

A palindrome is a number or a text phrase that reads the same backwards as forwards. For example, each of the following five-digit integers is a palindrome: 12321,55555,45554 and \(11611 .\) Write a program that reads in a five-digit integer and determines whether it is a palindrome. [Hint: Use the division and modulus operators to separate the number into its individual digits.]

Short answer

Extract digits using division and modulus; compare for equality in reverse order to check palindrome.

Step-by-step solution

Understand the problem

We need to determine if a given five-digit integer reads the same forwards and backwards. This means the first digit should match the last, and the second digit should match the second last.

Extract the digits

Given a five-digit number, extract each digit. For number 12345, the digits are: 1, 2, 3, 4, and 5. We can achieve this by using the division and modulus operators.

Use division and modulus

Let's say the number is represented by the variable 'num', first digit: \( d_1 = \frac{num}{10000} \); second digit: \( d_2 = \frac{(num/1000) \% 10}{1} \); middle digit: \( d_3 = \frac{(num/100) \% 10}{1} \); second last digit: \( d_4 = \frac{(num/10) \% 10}{1} \); last digit: \( d_5 = num \% 10 \).

Check for palindrome condition

A number is a palindrome if the first digit equals the last (\(d_1 = d_5\)) and the second digit equals the second-last (\(d_2 = d_4\)).

Implement the logic in code

Write a simple program: ``` num = int(input("Enter a five-digit integer: ")) d1 = num // 10000 d2 = (num // 1000) % 10 d3 = (num // 100) % 10 d4 = (num // 10) % 10 d5 = num % 10 if d1 == d5 and d2 == d4: print("The number is a palindrome.") else: print("The number is not a palindrome.") ```

Test the program

Verify the program by entering numbers such as 12321 and 45554 to see if it correctly identifies them as palindromes and entering numbers like 12345 to confirm it identifies them as not palindromes.

Key concepts

Palindrome Detection

Palindrome detection revolves around identifying if a sequence of numbers or characters reads the same forwards and backwards. In C++ programming, this concept is often applied to strings and numbers alike. To detect a palindrome, we compare the sequence using specific indexes to ensure symmetry from the start to end.

In the context of a five-digit integer, this involves checking if the first digit equals the last digit and the second digit shares the same value as the fourth one. By confirming these two conditions, we successfully determine if the integer is a palindrome. This symmetry makes palindromes unique and intriguing for programmers and mathematicians alike.

Modulus Operator

The modulus operator, denoted by the symbol `%`, is a fundamental tool in C++ programming for many tasks, especially when dealing with digits of a number. It computes the remainder of division and is particularly useful for extracting digits from an integer.

Let's take an example: when you perform `456 % 10`, the result is 6. This operation effectively helps us obtain the last digit of a number. By continuously dividing a number by 10 and taking its modulus, you can peel off digits from the end towards the start.

In the context of palindrome detection, the modulus operator helps isolate individual digits, allowing us to compare them across the number to assess symmetry.

Division Operator

In C++ programming, the division operator `/` is used just as one would expect in mathematics, to divide numbers. However, when used with integers, it performs integer division, meaning it only returns the whole number quotient, excluding the remainder.

For instance, performing `456 / 100` yields 4. In our palindrome detection program, this characteristic allows us to chop off digits starting from the last, moving them closer to the front digit by digit, when combined with modulus.

This interaction between modulus and division operators is essential to dissect a five-digit number into individual digits, like extracting the second digit by dividing the number by 1000, and further reducing it using modulus to skip digits not of interest. The cooperation of both operators simplifies the breakdown of numbers into quantifiable parts.

Five-Digit Integer

A five-digit integer is any number ranging from 10000 to 99999. These numbers hold significant educational value when studying concepts like palindrome detection in programming due to their structured length. Each digit can be systematically accessed and evaluated to reveal patterns, such as palindromes.

In solving palindrome problems, leveraging a five-digit integer provides a clear scope to operate within. It limits possibilities, making it computationally simpler compared to longer lengths, helping students focus on core programming logic and arithmetic rather than extensive data handling.

When working with a five-digit integer, it is crucial to have methods in place to accurately decompose and analyze each part, as seen with the division and modulus operators.

Programming Logic

Programming logic forms the backbone of creating efficient and accurate software solutions. It involves step-by-step processes and decision-making to ensure that a program functions as intended. When working with palindromes in C++, programming logic dictates the flow and structure of the solution.

For example, the logic here requires reading an integer, breaking it down using arithmetic operations, and comparing the resultant digits to establish a palindrome condition. A key step is deciding which operations best extract individual digits for comparison. The hands-on application of logical thinking allows developers to smoothly follow through with conditions and constraints effectively and logically.

Effective programming logic also means testing and verifying your solution with diverse inputs to check its robustness. This ensures that your code adheres strictly to logical accuracy and performs well under different conditions.