What are the prime factors of 189?

Steps to Solve

1. Determine if 189 is divisible by the smallest prime, 2

Since 189 is an odd number, it is not divisible by 2.

2. Check divisibility by the next prime, 3

To check if 189 is divisible by 3, sum the digits: $1 + 8 + 9 = 18$. Since 18 is divisible by 3, this means that 189 is divisible by 3.

3. Divide by 3 to find a factor

Now, divide 189 by 3: $$ 189 \div 3 = 63 $$

4. Factor 63 further

Next, we check if 63 can be factored further. Check 3 again, since summed digits of 63 are $6 + 3 = 9$, which is also divisible by 3. So, divide 63 by 3: $$ 63 \div 3 = 21 $$

5. Factor 21 further

Next, check 21 for divisibility by 3: The digit sum of 21 is $2 + 1 = 3$. Since 3 is divisible by 3, we divide: $$ 21 \div 3 = 7 $$

6. Identify the last factor

Now we are left with 7, which is a prime number.

7. Compile all prime factors

Now we can compile all the prime factors we found: The prime factorization of 189 is: $$ 3 \times 3 \times 3 \times 7 \text{ or } 3^3 \times 7 $$