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

The fastest way is to start dividing the number by the smallest prime number (2) and continue with the next primes until you’re left with only prime numbers.

For example, the prime factorization of 54 is:

$$ 2 \times 3 \times 3 \times 3 = 54 $$

Step-by-step: Prime Factors of 54

  1. Start with 54

  2. Divide by 2: $$ \frac{54}{2} = 27 $$ Add 2 to the factor list.

  3. Try 2 again: $$ \frac{27}{2} = 13.5 \quad \text{(not a whole number)} $$

    Try the next prime.

  4. Try 3: $$ \frac{27}{3} = 9 $$ Add 3 to the factor list.

  5. Try 3 again: $$ \frac{9}{3} = 3 $$ Add another 3.

  6. Try 3 again: $$ \frac{3}{3} = 1 $$ Add another 3.

You’re done.

Prime Factor List

The prime factors of 54 are:

$$ 2, 3, 3, 3 $$

Or using exponents:

$$ 2 \times 3^3 $$

Summary

To find the prime factors of a number:

  • Divide repeatedly by the smallest possible prime.
  • Add each prime factor to your list.
  • Stop when you’re left with 1.

So, the prime factorization of 54 is:

$$ 2 \times 3^3 $$

And to confirm:

$$ 2 \times 3 \times 3 \times 3 = 54 $$

✔️ Correct!