Introduction

When it comes to writing a program that prints numbers from 1 to N which are not divisible by 3 and 7 simultaneously, understanding the logic is key. This article will explore the different methods to achieve this, including algorithmic approaches and implementation in the Python programming language. We will also discuss how to optimize and structure this code for better performance and readability.

Understanding the Problem

The problem statement asks us to write a program that prints numbers from 1 to N which are not divisible by both 3 and 7 at the same time. This implies that a number can be either divisible by 3 or 7, but not both. Essentially, we are dealing with numbers in the range [1, N] that are not multiples of 21 (since 21 is the least common multiple of 3 and 7).

Algorithmic Approach

To solve this problem, we can use a simple algorithm that iterates through the range from 1 to N, checks if a number is not divisible by both 3 and 7 simultaneously, and prints it if the condition is met.

Version 1: Using Modulo Operator

In this version, we will use the modulo operator to check divisibility. The modulo operator (%) returns the remainder of the division of two numbers. If the remainder of a number divided by 21 is not zero, the number is not divisible by both 3 and 7.

def NOT3AND7(N):
    Z  []
    for i in range(1, N 1):
        if i % 21 ! 0:
            (i)
    return Z

This function takes an integer N as input and returns a list of numbers from 1 to N that are not divisible by both 3 and 7 simultaneously.

Version 2: Using Logical Operators

Another way to solve this problem is by using logical operators. We can check if a number is not divisible by 3 and not divisible by 7 using the logical AND operator (and).

def NOT3OR7(N):
    Z  []
    for i in range(1, N 1):
        if i % 3 ! 0 and i % 7 ! 0:
            (i)
    return Z

This function returns a list of numbers from 1 to N that are not divisible by 3 and 7.

Implementation in Python

To implement the above versions, let's use Python syntax. Here’s the code for the first version that checks if a number is not divisible by 21.

def NOT3AND7(N):
    Z  []
    for i in range(1, N 1):
        if i % 21 ! 0:
            (i)
    return Z
# Example usage
N  int(input("Enter a number: "))
result  NOT3AND7(N)
print(result)

The second version is implemented as follows:

def NOT3OR7(N):
    Z  []
    for i in range(1, N 1):
        if i % 3 ! 0 and i % 7 ! 0:
            (i)
    return Z
# Example usage
N  int(input("Enter a number: "))
result  NOT3OR7(N)
print(result)

Optimizing the Code

To optimize the code, we can avoid creating an intermediate list and instead print the numbers directly. This reduces memory usage and makes the code more efficient.

N  int(input("Enter a number: "))
# Version 1: Using modulo operator
for i in range(1, N 1):
    if i % 21 ! 0:
        print(i, end' ')  # prints the numbers separated by a space
print()  # prints a new line at the end
# Version 2: Using logical operators
for i in range(1, N 1):
    if i % 3 ! 0 and i % 7 ! 0:
        print(i, end' ')  # prints the numbers separated by a space
print()  # prints a new line at the end

Conclusion

Writing a program that prints numbers from 1 to N which are not divisible by 3 and 7 simultaneously is a common algorithmic problem. By using modular arithmetic, logical operators, and Python syntax, we can solve this problem effectively. The key is to understand the conditions and apply them correctly to achieve the desired result.

Whether you choose to use a list to store the results or directly output the numbers, both versions of the code are efficient and straightforward. The choice between them might depend on whether you are optimizing for memory usage (using a list) or for direct output without additional storage (printing directly).