Prime factorizations Mathematics

Prime factorizations

🔢 How to factor a number manually? (2 ways)

If you don’t have a calculator, you can use one of two popular methods to factorize a number into prime factors. Both of them lead to the same result, which is confirmed by the Fundamental Theorem of Arithmetic.

Method 1: Column division method (Classic)

This method is taught in most schools. You draw a vertical line: write the number on the left, and its smallest prime divisor on the right. The process continues until there is one left on the left side.

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Example for the number 60:

60 2
30 2
15 3
5 5
1

Result: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5 (Canonical expansion)

Method 2: “Multiplier Tree” Method (Visual)

This method is great for visual learners. You split a number into any two factors, then split each branch further until all that is left at the ends are primes.

Example for the number 48:

  1. 48 is divided into 6 × 8.
  2. 6 is divided into 2 × 3 (both are simple, branches are closed).
  3. split 8 into 2 × 4.
  4. 4 is divided into 2 × 2.

Collect all the “ends” of branches: 2, 3, 2, 2, 2.

Total: 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3 (Canonical expansion)

📘 Cheat sheet: Signs of divisibility

To quickly expand a number, you need to know what it is divisible by. Knowing these rules will save you a lot of time:

Divisor Rule Example
2 The number ends with 0, 2, 4, 6, 8. 126
3 The sum of the digits is divisible by 3. 123 (1+2+3=6)
5 The number ends with 0 or 5. 135
9 The sum of the digits is divisible by 9. 72 (7+2=9)
10 The number ends with 0. 570

⭐ Table of prime numbers (up to 100)

Primes are the “atoms” of mathematics. Decomposition always begins with checking divisibility by them.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Interesting fact: 2 is the only even prime number. All other prime numbers are odd.

🚀 Why is this necessary? (NOD and NOK)

Prime factorization is a key technique for working with fractions and large numbers.

  • GCD (Greatest Common Divisor): We take only common factors with the smallest degree.
  • LCM (Least Common Multiple): Take all encountered factors with the highest degree.

❓ Frequently asked questions (FAQ)

🔹 Is 1 a prime number?

No. Unit (1) is neither a prime nor a composite number. It has only one divisor (itself), and prime numbers must have exactly two divisors (1 and the number itself).

🔹 What is the smallest prime number?

The smallest prime number is 2. It is also the only even prime number.

🔹 What is “Canonical Decomposition”?

This is an entry where the factors are ordered in ascending order and grouped into powers. For example, instead of 2 × 2 × 2 × 3 we write 2³ × 3. Our calculator gives the answer exactly in this form.

🔹 How to find GCD and LCM through decomposition?

To find the gcd, multiply the common prime factors with minimum powers. For LCM, multiply all encountered prime factors with maximum powers.

Conclusion

The ability to factor a number into prime factors is the foundation of mathematics. Use the Long Division method for accuracy or the Multiplier Tree method for clarity. By using divisibility tests and knowing the basic prime numbers, you can quickly find GCD and LCM without even using an online calculator!

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