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Python Program to print all the numbers divisible by 3 and 5 for a given number

Python Program to Print all Prime numbers in an Interval

Last Updated : 21 Feb, 2025

The task of printing all prime numbers in an interval in Python involves taking two input values representing a range [x, y] and finding all prime numbers within that range. A prime number is a natural number greater than 1 that is divisible only by 1 and itself. For example, in the interval [2, 7], the prime numbers are [2, 3, 5, 7].

Using sieve of eratosthenes

Sieve of Eratosthenes is one of the most efficient algorithms to find all prime numbers in a given range, especially for large intervals. It works by creating a boolean array where each index represents a number, and values are marked as True for prime and False for non-prime. The algorithm eliminates multiples of each prime starting from 2.

Python

x, y = 2, 7  # define the range [x, y]  # create a boolean list to mark primes, assume all are prime initially primes = [True] * (y + 1)  # 0 and 1 are not prime primes[0], primes[1] = False, False  # Sieve of Eratosthenes to mark non-primes for i in range(2, int(y ** 0.5) + 1):     if primes[i]:         for j in range(i * i, y + 1, i):             primes[j] = False  # collect primes in the range [x, y] res = [i for i in range(x, y + 1) if primes[i]] print(res if res else "No")

Output

[2, 3, 5, 7]

Explanation: It creates a boolean list assuming all numbers are prime. Starting from 2 to √y, it marks multiples of each prime as False i.e., non-prime. After this, indices marked as True represent prime numbers in the range.

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Using trial division

Trial Division Method involves checking each number in the interval individually for primality. For each number, we test divisibility from 2 to √n. If a number is divisible by any value in this range, it is not prime. This method is simple and more efficient than the naive approach, especially for small intervals.

Python

x, y = 2, 7  # define range [x, y]  res = []  # list to store primes  for n in range(x, y + 1):     if n <= 1:         continue  # skip non-primes <= 1      is_prime = True  # assume n is prime     for i in range(2, int(n ** 0.5) + 1):         if n % i == 0:             is_prime = False  # n is not prime             break      if is_prime:         res.append(n)  # add prime number  print(res if res else "No")

Output

[2, 3, 5, 7]

Explanation: For each number n in the range, it checks if n is greater than 1. It then assumes n is prime and checks divisibility from 2 to √n. If n is divisible by any number in this range, it is marked as non-prime. If no divisors are found, n is added to the result list.

Using sympy library

SymPy Library provides a built-in function called primerange(x, y) to generate prime numbers in a given range efficiently. It is the most convenient and requires minimal coding effort. Internally, SymPy uses optimized algorithms, including the sieve method. This is ideal when you need a quick and accurate solution without implementing the logic manually.

Python

from sympy import primerange   x, y = 2, 7  # range [x, y]  primes = list(primerange(x, y + 1)) print(primes if primes else "No")

Output

[2, 3, 5, 7]

Explanation: sympy module, which generates prime numbers in a given range. The range is defined as [x, y] and primerange(x, y + 1) returns an iterator of prime numbers from x to y. The result is converted into a list and stored in primes.

Using naive approach

Naive Approach checks for primality by dividing each number from 2 to n//2. If any divisor is found, the number is not prime. This method is the least efficient and suitable only for small intervals . It is primarily used for understanding the basic concept of prime number checking.

Python

x, y = 2, 7  # range [x, y] res = []  # list to store primes  for i in range(x, y + 1):     if i <= 1:         continue  # skip non-primes <= 1     for j in range(2, i // 2 + 1):         if i % j == 0:             break  # not a prime     else:         res.append(i)  # prime found  print(res if res else "No")

Output

[2, 3, 5, 7]

Explanation: It loops through each number i from x to y and checks if i is a prime. If i is less than or equal to 1, it is skipped because primes are greater than 1. For each number i, it divides i by all numbers from 2 to i // 2. If any divisor is found, the loop breaks, indicating i is not prime. If no divisor is found, the else block runs, appending i to the result list.

Python Program to print all the numbers divisible by 3 and 5 for a given number

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