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 45 is:

$$ 3 \times 3 \times 5 = 45 $$

Step-by-step: Prime Factors of 45

  1. Start with 45

  2. Divide by 2: $$ \frac{45}{2} = 22.5 \quad \text{(not a whole number)} $$

    Move to the next prime.

  3. Try 3: $$ \frac{45}{3} = 15 $$ Add 3 to the factor list.

  4. Divide 15 by 3 again: $$ \frac{15}{3} = 5 $$ Add another 3.

  5. Try 3 again: $$ \frac{5}{3} \approx 1.67 \quad \text{(not a whole number)} $$

    Try the next prime.

  6. Try 5: $$ \frac{5}{5} = 1 $$ Add 5. You’re done.

Prime Factor List

The prime factors of 45 are:

$$ 3, 3, 5 $$

Or written with exponents:

$$ 3^2 \times 5 $$

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 45 is:

$$ 3^2 \times 5 $$

And to confirm:

$$ 3 \times 3 \times 5 = 45 $$

✔️ Correct!