Prime Number in Python: 7 Easy Methods with Code and Explanation

How do you check if a number is prime in Python—without using guesswork?It’s a common problem for beginners, but once you know the logic, it’s easier than it sounds.

A prime number in Python is just a number greater than 1 that’s only divisible by 1 and itself. If you're searching for a prime number program in Python or trying to understand the logic behind prime number code in Python, this guide will help. You'll learn to write a clean, working program for prime number in Python using simple loops and conditions.

We’ll also explain what makes a number prime, and how to optimize your code. Plus, if you’re aiming to build your Python basics for real-world use, check out our Data Science Courses and Machine Learning Courses for hands-on learning with logic-based problems.

What are the Different Methods to Check Prime Numbers in Python?

1. Python Program to Check Prime Number Using for-else statement

def is_prime(number):
    """
    Check if a given number is prime using for-else statement.
    
    Args:
        number (int): The number to be checked for primality.
    
    Returns:
        bool: True if the number is prime, False otherwise.
    """
    # Handle special cases
    if number < 2:
        return False
    
    # Check for divisibility from 2 to the square root of the number
    for divisor in range(2, int(number**0.5) + 1):
        # If number is divisible by any divisor, it's not prime
        if number % divisor == 0:
            return False
    else:
        # This block executes only if no divisors are found
        return True

# Test the function with different numbers
test_numbers = [2, 3, 4, 5, 7, 11, 12, 17, 20, 29]

print("Prime Number Checker Results:")
for num in test_numbers:
    result = is_prime(num)
    print(f"{num} is {'prime' if result else 'not prime'}")

Output:

Prime Number Checker Results:

2 is prime

3 is prime

4 is not prime

5 is prime

7 is prime

11 is prime

12 is not prime

17 is prime

20 is not prime

29 is prime

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Explanation of Code

1. Function Definition:
2. • We define is_prime() that takes a number as input
   • Handles special cases like numbers less than 2
   • Uses a for loop to check divisibility
2. Prime Checking Logic:
3. • Checks divisibility from 2 to the square root of the number
   • The else block of the for loop is key - it only runs if NO divisors are found
   • Returns True if no divisors are found, indicating a prime number
3. Optimization:
4. • We only check up to the square root of the number to improve efficiency
   • This reduces unnecessary iterations

Key Points About the For-Else Mechanism

• The else block executes only if the for loop completes without encountering a break
• In our prime number check, this means no divisors were found
• This provides a clean, Pythonic way to check primality

Related Reads:

• Python if-else statement
• Conditional Statement in Python

2. Program to Check Prime Number in Python Using Flag Variable

def is_prime(number):
    # Check if the number is less than 2
    # Numbers less than 2 are not prime
    if number < 2:
        return False
    
    # Initialize the flag as True
    # We assume the number is prime until proven otherwise
    is_prime_flag = True
    
    # Check for divisors from 2 to the square root of the number
    # We only need to check up to square root for efficiency
    for i in range(2, int(number**0.5) + 1):
        # If the number is divisible by any value, it's not prime
        if number % i == 0:
            # Set the flag to False
            is_prime_flag = False
            # Exit the loop as soon as a divisor is found
            break
    
    # Return the final state of the flag
    return is_prime_flag

# Test the function with different numbers
test_numbers = [2, 3, 4, 5, 7, 10, 11, 13, 17, 20]

# Print primality of each number
for num in test_numbers:
    print(f"{num} is {'prime' if is_prime(num) else 'not prime'}")

Output:

2 is prime

3 is prime

4 is not prime

5 is prime

7 is prime

10 is not prime

11 is prime

13 is prime

17 is prime

20 is not prime

Explanation of Code

Initial Check

• First, we check if the number is less than 2
• Numbers less than 2 are not considered prime
• If the number is less than 2, we immediately return False

Flag Initialization

• We start by setting is_prime_flag to True
• This assumes the number is prime until proven otherwise

Divisibility Check

• We iterate from 2 to the square root of the number
• We only need to check up to the square root for efficiency
• If any number divides the input number without a remainder, it's not prime

Flag Modification

• If a divisor is found, we set is_prime_flag to False
• We use break to exit the loop as soon as a divisor is found

Result Return

• The function returns the final state of the flag
• True means the number is prime
• False means the number is not prime

Key Advantages of Flag Variable

• Simple and easy to understand
• Efficient for small to medium-sized numbers
• Provides a clear boolean result
• Stops checking as soon as a divisor is found

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3. Prime Number Program in Python Using Break and Continue Method

def is_prime(number):
    # Handle edge cases
    if number < 2:
        return False
    
    # Check for divisibility from 2 to square root of the number
    for i in range(2, int(number**0.5) + 1):
        # If number is divisible by any value, it's not prime
        if number % i == 0:
            # Break the loop as soon as a divisor is found
            break
        
        # If no divisor found, continue checking
        continue
    else:
        # If loop completes without finding divisors, number is prime
        return True
    
    # If loop breaks due to finding a divisor, number is not prime
    return False

# Test the function with various numbers
test_numbers = [2, 3, 4, 5, 7, 10, 11, 13, 17, 20]

print("Prime Number Checking Results:")
for num in test_numbers:
    result = is_prime(num)
    print(f"{num} is {'prime' if result else 'not prime'}")

Output:

Prime Number Checking Results:

2 is prime

3 is prime

4 is not prime

5 is prime

7 is prime

10 is not prime

11 is prime

13 is prime

17 is prime

20 is not prime

Explanation of Code

1. Function Definition:
2. • is_prime(number) takes a number as input and returns True if it's prime, False otherwise.
2. Edge Case Handling:
3. • If the number is less than 2, it returns False because prime numbers start from 2.
3. Divisibility Check:
4. • Loops from 2 to the square root of the number (optimization technique).
   • Checks if the number is divisible by any value in this range.
4. Break Statement:
5. • If a divisor is found, break immediately exits the loop.
   • This happens when number % i == 0 (number is perfectly divisible).
5. Continue Statement:
6. • continue is used to skip to the next iteration if no divisor is found.
   • In this implementation, it doesn't significantly change the logic but demonstrates its usage.
6. Else Clause with For Loop:
7. • If the loop completes without finding divisors, the else block is executed.
   • This indicates the number is prime.

Related Read:

• Break, Pass, and Continue Statement in Python
• Break Statement in Python

4. Python Program for Prime Number Using While Loop

def is_prime(number):
    # Check if the number is less than 2 (not prime)
    if number < 2:
        return False
    
    # Initialize the divisor
    divisor = 2
    
    # Use while loop to check divisibility
    while divisor * divisor <= number:
        # Check if the number is divisible by the current divisor
        if number % divisor == 0:
            # If divisible, it's not a prime number
            return False
        
        # Increment the divisor
        divisor += 1
    
    # If no divisors found, the number is prime
    return True

# Test the function
test_numbers = [2, 3, 4, 5, 7, 10, 11, 13, 17, 20]

# Print prime status for each number
for num in test_numbers:
    if is_prime(num):
        print(f"{num} is a prime number")
    else:
        print(f"{num} is not a prime number")

Output:

2 is a prime number

3 is a prime number

4 is not a prime number

5 is a prime number

7 is a prime number

10 is not a prime number

11 is a prime number

13 is a prime number

17 is a prime number

20 is not a prime number

Explanation of Code

Function Definition

• We define a function is_prime() that takes a number as input
• This function will return True if the number is prime, False otherwise

Initial Check

• First, we check if the number is less than 2
• Numbers less than 2 are not considered prime
• If the number is less than 2, we immediately return False

While Loop Mechanism

• We start with a divisor of 2 (the smallest prime number)
• The while loop continues as long as the square of the divisor is less than or equal to the number
• This optimization reduces unnecessary iterations

Divisibility Check

• Inside the loop, we check if the number is divisible by the current divisor
• If the remainder is 0, the number is not prime
• We return False as soon as a divisor is found

Divisor Increment

• If no divisibility is found, we increment the divisor
• This allows us to check the next potential divisor

Prime Confirmation

• If the loop completes without finding any divisors, the number is prime
• We return True

Related Read

• Python While Loop

5. Python Code for Prime Number Using Math Module

import math

def is_prime_using_math(number):
    """
    Check if a given number is prime using the math module.
    
    A prime number is a natural number greater than 1 that is only divisible by 1 and itself.
    This method uses mathematical properties to efficiently check primality.
    """
    # Handle special cases
    if number <= 1:
        return False
    
    # 2 is the only even prime number
    if number == 2:
        return True
    
    # Even numbers greater than 2 are not prime
    if number % 2 == 0:
        return False
    
    # Use math.isqrt to find the integer square root efficiently
    # We only need to check divisors up to the square root of the number
    sqrt_number = math.isqrt(number)
    
    # Check for divisibility by odd numbers up to the square root
    for divisor in range(3, sqrt_number + 1, 2):
        # If the number is divisible by any odd number, it's not prime
        if number % divisor == 0:
            return False
    
    # If no divisors are found, the number is prime
    return True

# Example usage and demonstration
def demonstrate_prime_checking():
    # List of numbers to test
    test_numbers = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 19, 20, 29]
    
    print("Prime Number Checking Results:")
    print("-" * 30)
    
    for num in test_numbers:
        result = is_prime_using_math(num)
        print(f"{num} is {'prime' if result else 'not prime'}")

# Run the demonstration
demonstrate_prime_checking()

Output:

Prime Number Checking Results:

------------------------------

1 is not prime

2 is prime

3 is prime

4 is not prime

5 is prime

6 is not prime

7 is prime

8 is not prime

9 is not prime

10 is not prime

11 is prime

12 is not prime

13 is prime

17 is prime

19 is prime

20 is not prime

29 is prime

Explanation

• Import the math module
• Define a function is_prime_using_math()
• Handle special cases (numbers ≤ 1, 2, even numbers)
• Calculate the integer square root using math.isqrt()
• Check divisibility by odd numbers up to the square root
• Return True if no divisors are found, False otherwise

6. Prime Number Code in Python Using Recursion Method

def is_prime_recursive(n, divisor=None):
    """
    Check if a number is prime using recursive approach
    
    Args:
    n (int): The number to check for primality
    divisor (int, optional): The current divisor to check against. 
                             Defaults to None for initial call.
    
    Returns:
    bool: True if the number is prime, False otherwise
    """
    # Handle special cases
    if n <= 1:
        return False
    
    # If no divisor specified, start with square root of n
    if divisor is None:
        divisor = int(n ** 0.5)
    
    # Base cases
    # If we've checked all potential divisors successfully, the number is prime
    if divisor == 1:
        return True
    
    # If the number is divisible, it's not prime
    if n % divisor == 0:
        return False
    
    # Recursively check with the next lower divisor
    return is_prime_recursive(n, divisor - 1)

# Test the function with different numbers
test_numbers = [2, 3, 4, 7, 10, 17, 20, 29]

print("Prime Number Recursion Test Results:")
for num in test_numbers:
    result = is_prime_recursive(num)
    print(f"{num} is {'prime' if result else 'not prime'}")

Output:

Prime Number Recursion Test Results:

2 is prime

3 is prime

4 is not prime

7 is prime

10 is not prime

17 is prime

20 is not prime

29 is prime

Explanation

The recursive approach to checking prime numbers involves systematically testing potential divisors from the square root of the number down to 1. Instead of using a loop, we use recursive function calls to check divisibility.

Key Recursive Strategy:

1. Start with the divisor as the square root of the number
2. Recursively check divisibility by reducing the divisor
3. Use base cases to determine primality

Code Breakdown:

• The function is_prime_recursive() takes two parameters:
• • n: The number to check for primality
  • divisor: Optional parameter to track current divisor (defaults to None)

Recursion Logic:

• Special case handling for numbers ≤ 1 (not prime)
• If no divisor is specified, start with square root of n
• Base cases:
• • If divisor reaches 1, number is prime
  • If number is divisible by current divisor, it's not prime
• Recursive call with reduced divisor

Step-by-Step Explanation:

1. When checking if a number is prime, we start from its square root
2. We recursively divide the number by decreasing divisors
3. If any divisor divides the number exactly, it's not prime
4. If we reach 1 without finding any divisors, the number is prime

Related Read:

• Recursion in Python
• Python Recursive Function

7. Prime Number Code in Python Using sympy.isprime()

pip install sympy
# Import the isprime function from sympy library
from sympy import isprime

# Function to demonstrate prime number checking
def check_prime_numbers():
    # Test various numbers to demonstrate isprime() functionality
    test_numbers = [2, 7, 10, 17, 20, 29, 100, 997]
    
    # Iterate through each number and check its primality
    for number in test_numbers:
        # Use isprime() to determine if the number is prime
        if isprime(number):
            print(f"{number} is a prime number.")
        else:
            print(f"{number} is not a prime number.")

# Call the function to run the demonstration
check_prime_numbers()

Output:

2 is a prime number.

7 is a prime number.

10 is not a prime number.

17 is a prime number.

20 is not a prime number.

29 is a prime number.

100 is not a prime number.

997 is a prime number.

Explanation of Code

Import the Method:

• We import isprime() from the sympy library
• This gives us direct access to an optimized prime checking function

Function Creation:

• We create a function check_prime_numbers() to demonstrate the method
• This allows us to test multiple numbers easily

Number Selection:

• We create a list test_numbers with various integers
• The list includes prime and non-prime numbers to show the method's versatility

Primality Testing:

• We use a for loop to iterate through each number
• isprime() returns True for prime numbers, False otherwise
• We use an if-else statement to print whether each number is prime

Method Advantages:

• Works efficiently for small and large numbers
• No need to implement complex prime-checking algorithms manually
• Highly reliable and mathematically accurate

8. Bonus: Large Number Primarily Test

# Large number primality test
large_number = 104729
print(f"{large_number} is prime: {isprime(large_number)}")
# Interactive user input
def interactive_prime_check():
    try:
        number = int(input("Enter a number to check if it's prime: "))
        if isprime(number):
            print(f"{number} is a prime number!")
        else:
            print(f"{number} is not a prime number.")
    except ValueError:
        print("Please enter a valid integer.")

Output:

104729 is prime: True

Performance Analysis of Different Methods Using Time Complexity and Space Complexity

Note: The complexity of sympy.isprime() varies based on its internal optimizations for small and large numbers.

Related Read:

• Time and Space Complexity

MCQs on Prime Number in Python

1. Which of the following is a prime number?

a) 4

b) 6

c) 7

d) 9

2. What is the output of the code: print(5 % 2)?

a) 2

b) 0

c) 1

d) 5

3. What does the following function check?

def is_prime(n):  
    for i in range(2, n):  
        if n % i == 0:  
            return False  
    return True  

a) Checks if a number is even

b) Checks if a number is composite

c) Checks if a number is prime

d) Checks if a number is odd

4. Which loop is best suited for checking if a number is prime?

a) `while` loop

b) `for` loop

c) `do-while` loop

d) `None`

5. What will be the output of:

def check_prime(n):  
    for i in range(2, n):  
        if n % i == 0:  
            print("Not Prime")  
            break  
    else:  
        print("Prime")  
check_prime(11)  

a) Prime

b) Not Prime

c) Error

d) Nothing

6. Which Python keyword is used to exit a loop early in a prime number check?

a) `continue`

b) `exit`

c) `return`

d) `break`

7. What is the time complexity of a basic prime number program in Python using a loop up to n?

a) O(1)

b) O(log n)

c) O(n)

d) O(n²)

8. What change improves the performance of this basic check?

for i in range(2, n):  
    if n % i == 0:  

a) Loop till n//2

b) Loop till n-1

c) Loop till sqrt(n)

d) Loop from 0 to n

9. You need a function that returns a list of all prime numbers from 1 to 100. What Python construct is best?

a) `lambda`

b) `while` loop

c) `for` loop with list

d) `try-except`

10. A user inputs a number. You want to check if it’s prime and print the result. Which line is correct?

a) `if is_prime(num): print("Prime")`

b) `if prime(num): print("Yes")`

c) `if num == prime: print("True")`

d) `print(prime(num))`

11. You’re asked to optimize a prime number code for large inputs. What should you use?

a) Sieve of Eratosthenes

b) Factorial function

c) GCD algorithm

d) Binary search

Conclusion 

Understanding how to check for a Prime Number in Python is a fundamental exercise that sharpens your problem-solving and optimization skills. This tutorial has walked you through various methods, from the simple trial division to more efficient, optimized approaches. 

By building and refining a program for prime number in Python, you're not just solving a classic coding challenge, you're mastering core concepts like loops, conditionals, and computational efficiency. Keep practicing with these techniques, and you'll be well-prepared for more complex algorithmic problems in your programming journey. 

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FAQ’s

1. What is a prime number?  

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, it cannot be formed by multiplying two smaller natural numbers. For example, 5 is a prime number because its only divisors are 1 and 5. The number 6, however, is not prime (it is a composite number) because it can be divided by 2 and 3. Understanding this definition is the first step in creating a program for prime number in Python. 

2. Why are 0 and 1 not considered prime numbers?  

By definition, a prime number must be a natural number greater than 1. The number 1 is excluded because it only has one distinct positive divisor (itself), whereas a prime number must have exactly two. If 1 were considered prime, the fundamental theorem of arithmetic (which states that every integer greater than 1 is either a prime number itself or can be represented as a product of prime numbers) would no longer be unique. 0 is not prime because it can be divided by any non-zero integer. 

3. How can I find the prime number in Python?  

To determine if a number is a Prime Number in Python, you need to write code that checks if it is divisible by any integer other than 1 and itself. The most common method is trial division, where you create a loop that iterates from 2 up to the square root of the number. Inside the loop, you use the modulo operator (%) to check for divisibility. If any number in that range divides the input number evenly (i.e., the remainder is 0), then the number is not prime. If the loop completes without finding any divisors, the number is prime. 

4. What is the formula for finding prime numbers?  

There is no single, simple algebraic formula that can generate all prime numbers. Prime numbers are a subject of deep mathematical study, and their distribution is complex. Instead of a formula, programmers use algorithms to find them. For generating a list of primes up to a certain limit, the most famous algorithm is the Sieve of Eratosthenes. For checking a single number, the best approach is an optimized trial division, which is the basis for any good program for prime number in Python. 

5. What are the prime numbers from 0 to 100 in Python?  

The prime numbers between 0 and 100 are a classic set to generate when you are first learning to code. The complete list of these 25 prime numbers is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Writing a Python script to generate this list is an excellent beginner exercise. 

6. How to Check Prime Numbers in Python?  

There are many methods to check for a Prime Number in Python, each with different trade-offs in readability and efficiency. A simple approach is using a for loop with an if-else statement to check for divisors. You can also use a flag variable that gets toggled if a divisor is found. For more concise code, you can use a for-else loop, where the else block only runs if the loop completes without hitting a break. More advanced methods include using recursion or pre-built functions from libraries like SymPy. 

7. How to Find a Prime Number in Python with a Function? 

Using a function is the best practice for creating a reusable and clean program for prime number in Python. A well-designed function, say is_prime(n), would take an integer n as input and return a boolean value (True or False). Inside the function, you would encapsulate all the logic, including handling edge cases (like numbers less than 2) and implementing the optimized trial division loop. This makes your main code much more readable, as you can simply call is_prime(number) to get a clear answer. 

8. How to find a prime number in Python without a loop?  

While loops are the most intuitive way to implement the logic, you can check for a Prime Number in Python without writing an explicit loop by using more advanced techniques. Recursion is one method, where a function calls itself to perform the divisibility checks. A more practical and common approach is to use a pre-existing, highly optimized function from a library. For example, the SymPy library has a function sympy.isprime() that can determine if a number is prime very efficiently. 

9. How do you implement the Sieve of Eratosthenes in Python?  

The Sieve of Eratosthenes is an ancient and highly efficient algorithm for finding all prime numbers up to a specified limit. To implement it, you start by creating a boolean list (or array) representing all numbers from 0 to your limit, initially marking them all as potentially prime. You then start with the first prime, 2, and iterate through the list, marking all multiples of 2 as not prime. You then move to the next unmarked number, 3, and do the same. This process continues until you have iterated through the list, leaving only the prime numbers marked as true. 

10. What is the time complexity of a prime number program in Python? 

The time complexity of a program for prime number in Python depends on the algorithm used. A naive approach that checks for divisors from 2 up to n-1 has a time complexity of O(n). A significantly better and more common approach is to check for divisors only up to the square root of n, which improves the time complexity to O(√n). The Sieve of Eratosthenes algorithm, which generates all primes up to n, is even more efficient, with a time complexity of O(n log log n). 

11. What is the fastest way to check for a prime number in Python?  

The fastest method depends on the size of the number being checked. For most practical purposes and numbers up to a very large size, an optimized trial division is sufficient. This involves checking for divisibility only by 2 and then by odd numbers up to the square root of the number. For extremely large numbers, such as those used in cryptography, probabilistic primality tests like the Miller-Rabin test are much faster, as they can determine the likelihood of a number being prime with a very high degree of certainty. 

12. How to Generate Prime Numbers in Python?  

There are two main ways to generate a list of prime numbers. To generate primes within a finite range (e.g., all primes under 1,000), the most efficient method is to use an algorithm like the Sieve of Eratosthenes. To generate an infinite sequence of primes, you can create a generator function. This function would yield primes one at a time as they are found, which is very memory-efficient as it doesn't store the entire list in memory. 

13. What is a composite number?  

A composite number is a natural number greater than 1 that is not a prime number. In other words, it is a number that has at least one positive divisor other than 1 and itself. For example, 9 is a composite number because it can be divided by 3. When you write a program for prime number in Python, your logic is essentially trying to determine if a number is composite; if it's not, and it's greater than 1, then it must be prime. 

14. Why do we only need to check for divisors up to the square root of a number?  

This is a crucial optimization for any primality test. If a number n has a divisor d that is larger than its square root (√n), then n must also have another divisor c such that d * c = n. In this case, c must be smaller than the square root of n. Therefore, if n has any divisors, at least one of them must be less than or equal to its square root. This means we don't need to check any further, which dramatically reduces the number of iterations in our program for prime number in Python. 

15. What are twin primes, and how can you find them in Python?  

Twin primes are a pair of prime numbers that differ by exactly 2. For example, (3, 5), (5, 7), and (11, 13) are all twin prime pairs. To find them in Python, you can first generate a list of all prime numbers up to a certain limit. Then, you can iterate through that list and check if the difference between a prime and the next prime in the list is equal to 2. 

16. Are there any built-in Python libraries for working with prime numbers?  

Python's standard library does not have a dedicated, built-in function to check for primality. However, the math module provides helpful functions like math.isqrt() for efficient square root calculation, which is useful in a program for prime number in Python. For more advanced or high-performance primality testing, it is common to use third-party libraries like SymPy or gmpy2, which have highly optimized functions for this purpose. 

17. What are some real-world applications of prime numbers? 

Prime numbers are absolutely critical in the field of cryptography, which is the foundation of modern computer security. Public-key cryptography algorithms, such as RSA, rely on the mathematical properties of large prime numbers. The security of these systems is based on the fact that it is computationally very easy to multiply two large prime numbers together, but it is extremely difficult to do the reverse—that is, to factor the product back into its original prime components. 

18. How can I write a more "Pythonic" prime number checker?

A "Pythonic" approach often favors readability and conciseness. Instead of a complex loop with a flag, you could write a function that uses a generator expression with the all() function. For example, all(n % i != 0 for i in range(2, int(n**0.5) + 1)). This single line of code checks if all the potential divisions have a non-zero remainder. This is a more advanced but elegant way to structure your Prime Number in Python logic. 

19. How can I improve my problem-solving skills with prime number challenges?  

Prime numbers are a fantastic topic for honing your algorithmic thinking. You can start by implementing the basic trial division method, then move on to optimizing it. After that, challenge yourself to implement the Sieve of Eratosthenes. For a more structured learning path, you can explore the programming and algorithms courses offered by upGrad, which often include these types of classic problems and provide expert guidance on different optimization techniques. 

20. What is the main takeaway from learning to write a program for a prime number in Python?  

The main takeaway is not just about identifying a Prime Number in Python, but about learning the fundamentals of algorithmic thinking and optimization. The process of starting with a simple, brute-force solution and then refining it by applying mathematical properties (like checking only up to the square root) is a perfect example of the problem-solving mindset that is at the heart of computer science. It's a foundational exercise that teaches you how to write code that is not just correct, but also efficient. 

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