How can you find all the prime factors of a number quickly and easily?

The fastest way to find all the prime factors of a number is:

  1. Start with the smallest prime number (2).
  2. Check if it divides evenly.
  3. If it does, record it and divide again.
  4. If not, move to the next prime.

For example, for the number 44 the prime factorization is:

$$ 2 \times 2 \times 11 = 44 $$

Find Prime Factors of 44

Let’s break this down step-by-step:

  1. Start with 44.

  2. Try dividing by the first prime number, 2:

    $$ \frac{44}{2} = 22 $$

    ✅ Add 2 to your prime factors.

  3. Try 2 again:

    $$ \frac{22}{2} = 11 $$

    ✅ Add another 2 to your list.

  4. Try 2 again:

    $$ \frac{11}{2} = 5.5 \quad \text{(not a whole number)} $$

    ❌ Not divisible by 2.

  5. Try next primes: 3, 5, 7 — none divide 11.

  6. Try 11:

    $$ \frac{11}{11} = 1 $$

    ✅ Add 11 to your list.

At this point, you’re done.

Prime Factor List

So the prime factors of 44 are:

$$ 2 \times 2 \times 11 $$

To check:

$$ 2 \times 2 \times 11 = 4 \times 11 = 44 $$

✔️ Confirmed.

Summary

To find the prime factorization of a number:

  • Begin with the smallest prime.
  • Divide repeatedly.
  • Record each divisor.
  • Stop once the quotient is 1.

The prime factorization of 44 is:

$$ 2 \times 2 \times 11 $$

To move faster, memorize primes up to 20:

$$ 2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19 $$

This makes your search for divisors much quicker!