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 keep dividing until you’re left with a set of prime numbers whose product is your original number.

For example, the prime factors of 63 are:

$$ 3 \times 3 \times 7 = 63 $$

Step-by-step: Prime Factorization of 63

  1. Start with 63.

  2. Try dividing by 2: $$ \frac{63}{2} = 31.5 \quad \text{(not whole)} $$ Skip 2.

  3. Try 3: $$ \frac{63}{3} = 21 $$ Add 3 to the prime factor list.

  4. Divide 21 by 3: $$ \frac{21}{3} = 7 $$ Add another 3.

  5. Try 3 again: $$ \frac{7}{3} \approx 2.33 \quad \text{(not whole)} $$

  6. Try next primes: 4 (not prime), 5 (doesn’t divide), then 7: $$ \frac{7}{7} = 1 $$ Add 7.

You’re done.

Prime Factor List

The prime factors of 63 are:

$$ 3, 3, 7 $$

Or with exponents:

$$ 3^2 \times 7 $$

Summary

To find the prime factors:

  • Divide by the smallest possible prime number.
  • Repeat division by that prime until it’s no longer a factor.
  • Move to the next prime number.
  • Continue until the quotient is 1.

So, the prime factorization of 63 is:

$$ 3^2 \times 7 $$

To verify:

$$ 3 \times 3 \times 7 = 63 \quad \text{✔} $$