How Can You Find All the Prime Factors of a Number Quickly?
The fastest way to find all the prime factors of a number is to start with the first prime number ($2$), and continue dividing until the number is reduced entirely into prime numbers.
For example, for the number $84$, the prime factors are:
$$ 2 \times 2 \times 3 \times 7 $$
Step-by-Step Prime Factorization of 84
Let’s find the prime factors of $84$ step by step:
- $84 \div 2 = 42$ → Add $2$
- $42 \div 2 = 21$ → Add $2$
- $21 \div 2$ → Not divisible
- Move to $3$
- $21 \div 3 = 7$ → Add $3$
- $7 \div 3$ → Not divisible
- Test $4$ (not prime), skip
- Test $5$ → $7 \div 5$ → Not divisible
- Test $6$ (not prime), skip
- $7 \div 7 = 1$ → Add $7$ and stop
So the complete list of prime factors is:
$$ \boxed{2 \times 2 \times 3 \times 7} $$
Final Check
Multiply them together to verify:
$$ 2 \times 2 \times 3 \times 7 = 84 $$
✅ Correct!
Summary
To find the prime factors of a number:
- Start at $2$
- Keep dividing by each prime as long as it’s a factor
- Once it no longer divides evenly, move to the next prime
- Stop when the quotient is $1$
Knowing the first few prime numbers makes this process faster.
The prime factorization of 84 is:
$$ \boxed{2 \times 2 \times 3 \times 7} $$
Multiply all prime factors together to confirm the result.