what number is 399 divisible by

2 min read 25-12-2024

Finding the divisors of a number like 399 might seem daunting, but it's a straightforward process using a combination of math techniques and a little bit of cleverness. This article will guide you through several methods to determine all the numbers that 399 is divisible by. We'll explore prime factorization, divisibility rules, and even a bit of code to help you tackle similar problems in the future.

Understanding Divisibility

Before diving into the specifics of 399, let's quickly refresh the concept of divisibility. A number is divisible by another number if the result of their division is a whole number (no remainder). For example, 12 is divisible by 3 because 12/3 = 4.

Method 1: Prime Factorization

Prime factorization is a powerful technique to find all divisors of a number. It involves breaking down the number into its prime factors – numbers only divisible by 1 and themselves.

Finding the Prime Factors of 399

Let's find the prime factors of 399:

Start with small prime numbers: We can see that 399 is divisible by 3 (because the sum of its digits, 3 + 9 + 9 = 21, is divisible by 3).
Divide: 399 / 3 = 133
Continue: Now we need to factor 133. It's not divisible by 2, 3, 5, or 7. However, it is divisible by 7.
Final Factorization: 133 / 7 = 19. 19 is a prime number.

Therefore, the prime factorization of 399 is 3 x 7 x 19.

Finding Divisors from Prime Factors

To find all divisors, we consider all possible combinations of the prime factors:

1
3
7
19
3 x 7 = 21
3 x 19 = 57
7 x 19 = 133
3 x 7 x 19 = 399

Therefore, the numbers 399 is divisible by are 1, 3, 7, 19, 21, 57, 133, and 399.

Method 2: Divisibility Rules

Divisibility rules offer quick checks for divisibility by certain numbers. While not exhaustive, they can help identify some divisors rapidly.

Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. (3 + 9 + 9 = 21, which is divisible by 3)
Divisibility by 7: There isn't a simple rule for 7, but we already found that 399 is divisible by 7 in the prime factorization.
Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. (3 + 9 + 9 = 21, which is not divisible by 9)
Divisibility by 19: Again, no easy rule, but we know from prime factorization.

Method 3: Using a Computer Program

For larger numbers, a computer program can be helpful. Here's a simple Python script to find all divisors:

def find_divisors(num):
  divisors = []
  for i in range(1, num + 1):
    if num % i == 0:
      divisors.append(i)
  return divisors

print(find_divisors(399)) # Output: [1, 3, 7, 19, 21, 57, 133, 399]

This script iterates through numbers from 1 to 399 and checks for divisibility.

Conclusion

399 is divisible by 1, 3, 7, 19, 21, 57, 133, and 399. This article demonstrated several approaches—prime factorization, divisibility rules, and a simple computer program—to find these divisors. Understanding these methods empowers you to tackle similar divisibility problems efficiently, regardless of the number's size. Remember that prime factorization provides the most complete and systematic way to determine all divisors of any given number.