What is the prime factorization of 165?

Steps to Solve

1. Start with the smallest prime number
   Check if 2 is a factor of 165. Since 165 is odd, it is not divisible by 2.

2. Check the next prime number
   Now check if 3 is a factor of 165. To do this, add the digits of 165: $1 + 6 + 5 = 12$. Since 12 is divisible by 3, this means 165 is also divisible by 3.

3. Divide by 3
   Perform the division:
   $$ \frac{165}{3} = 55 $$
   So, 3 is one of the prime factors.

4. Factor the result (55)
   Now we need to find the prime factors of 55. Start with the smallest prime number again, which is 2. 55 is odd, so it is not divisible by 2.

5. Check the next prime number (3)
   Check if 3 is a factor of 55 by adding the digits: $5 + 5 = 10$. Since 10 is not divisible by 3, 55 is not divisible by 3.

6. Check the next prime number (5)
   Check if 5 is a factor of 55. The last digit of 55 is 5, which means it is divisible by 5.

7. Divide by 5
   Perform the division:
   $$ \frac{55}{5} = 11 $$
   So, 5 is another prime factor.

8. Check the result (11)
   Finally, we need to check if 11 is a prime number. The only divisors of 11 are 1 and 11 itself, making it a prime number.

9. Combine the prime factors
   The prime factors of 165 are 3, 5, and 11.