Check if product of first N natural numbers is divisible by their sum in Python

Suppose we have a number n. We have to check whether the product of (1*2*...*n) is divisible by (1+2+...+n) or not.

So, if the input is like num = 5, then the output will be True as (1*2*3*4*5) = 120 and (1+2+3+4+5) = 15, and 120 is divisible by 15.

Mathematical Approach

Instead of calculating the actual product and sum, we can use a mathematical property. The product of first n natural numbers is n! (factorial), and the sum is n*(n+1)/2. The key insight is:

• If num + 1 is prime, then the factorial is NOT divisible by the sum
• If num + 1 is not prime, then the factorial IS divisible by the sum

Algorithm Steps

To solve this, we will follow these steps −

• If num + 1 is prime, then
  • return False
• Return True

Implementation

Let us see the following implementation to get better understanding −

def isPrime(num):
    if num > 1:
        for i in range(2, num):
            if num % i == 0:
                return False
        return True
    return False

def solve(num):
    if isPrime(num + 1):
        return False
    return True

num = 5
print(f"For n = {num}:")
print(f"Product (factorial): {1*2*3*4*5}")
print(f"Sum: {1+2+3+4+5}")
print(f"Is divisible: {solve(num)}")

For n = 5:
Product (factorial): 120
Sum: 15
Is divisible: True

Testing Multiple Cases

Let's test with different values to verify our approach ?

def isPrime(num):
    if num > 1:
        for i in range(2, num):
            if num % i == 0:
                return False
        return True
    return False

def solve(num):
    if isPrime(num + 1):
        return False
    return True

# Test multiple cases
test_cases = [3, 4, 5, 6, 7]

for n in test_cases:
    result = solve(n)
    print(f"n = {n}, n+1 = {n+1}, is_prime({n+1}) = {isPrime(n+1)}, divisible = {result}")

n = 3, n+1 = 4, is_prime(4) = False, divisible = True
n = 4, n+1 = 5, is_prime(5) = True, divisible = False
n = 5, n+1 = 6, is_prime(6) = False, divisible = True
n = 6, n+1 = 7, is_prime(7) = True, divisible = False
n = 7, n+1 = 8, is_prime(8) = False, divisible = True

Why This Works

The mathematical reasoning behind this approach is based on properties of factorials and prime numbers. When n+1 is prime, it creates a specific relationship between the factorial and the sum that prevents divisibility.

Conclusion

This problem can be solved efficiently by checking if n+1 is prime instead of computing large factorials. If n+1 is prime, the product is not divisible by the sum; otherwise, it is divisible.