How Do You Find the Prime Factors of 75?

The fastest way to find all the prime factors of a number is to start with the smallest prime (2) and test divisibility. If divisible, divide and repeat. Continue with increasing prime numbers until the number is fully factored.

For example, for the number 75, the prime factors are:

$$ 3 \times 5 \times 5 $$

This means:

$$ 3 \times 5 \times 5 = 75 $$

Step-by-Step Prime Factorization of 75

Let’s find the prime factors of 75 using the step-by-step method:

  1. Start with the first prime number: $2$
    $75 \div 2 = 37.5$ → not divisible
  2. Try the next prime: $3$
    $75 \div 3 = 25$ → divisible
    Add $3$ to the prime factor list.
  3. Test $3$ again on $25$:
    $25 \div 3 \approx 8.33$ → not divisible
  4. Next prime is $5$:
    $25 \div 5 = 5$ → divisible
    Add $5$ to the prime factor list.
  5. Test $5$ again on $5$:
    $5 \div 5 = 1$ → divisible
    Add $5$ to the prime factor list.

You now have all the prime factors of 75:

$$ \boxed{3 \times 5 \times 5} $$

Final Check

Let’s confirm:

$$ 3 \times 5 \times 5 = 75 $$

✅ The product matches the original number.

Summary

To find the prime factors of a number:

  • Start with the smallest prime number ($2$)
  • If divisible, divide and repeat with the quotient
  • If not, move to the next prime number
  • Repeat until the quotient is $1$

Tip:

Memorize the first few prime numbers (up to $20$) to make this process faster.

For $75$, the prime factorization is:

$$ \boxed{3 \times 5 \times 5} $$