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 to start with the smallest prime number (2), and divide repeatedly, moving to the next prime only when necessary, until you’re left with all the prime numbers that multiply to the original number.

For example, the prime factors of 56 are:

$$ 2 \times 2 \times 2 \times 7 = 56 $$

Step-by-step: Prime Factorization of 56

  1. Start with 56

  2. Divide by 2: $$ \frac{56}{2} = 28 $$ Add 2 to the factor list.

  3. Divide 28 by 2: $$ \frac{28}{2} = 14 $$ Add another 2.

  4. Divide 14 by 2: $$ \frac{14}{2} = 7 $$ Add another 2.

  5. Try 2 again: $$ \frac{7}{2} = 3.5 \quad \text{(not whole)} $$

  6. Try the next primes: 3, 5 — no match.

  7. Try 7: $$ \frac{7}{7} = 1 $$ Add 7.

You’re done.

Prime Factor List

The prime factors of 56 are:

$$ 2, 2, 2, 7 $$

Or with exponents:

$$ 2^3 \times 7 $$

Summary

To find prime factors:

  • Begin dividing by the smallest prime number (2).
  • Continue dividing by the same prime until it no longer divides evenly.
  • Then move to the next prime.
  • Repeat until the quotient is 1.

So, the prime factorization of 56 is:

$$ 2^3 \times 7 $$

To check:

$$ 2 \times 2 \times 2 \times 7 = 56 \quad \text{✔} $$