Find the prime factorization of 90.

Answer

To find the prime factorization of the number 90, we will follow a systematic approach using the division method. Here are the steps:

### Step-by-Step Solution:

1. **Start with the number 90**:
We begin with the number we want to factorize, which is 90.

2. **Divide by the smallest prime number**:
The smallest prime number is 2. We check if 90 is divisible by 2.
\[ 90 \div 2 = 45 \]
Since 90 is even, it is divisible by 2. We write down 2 as one of the factors.

3. **Continue with the quotient (45)**:
Next, we take the quotient we obtained, which is 45, and check if it is divisible by 2.
Since 45 is odd, we move to the next smallest prime number, which is 3.
\[ 45 \div 3 = 15 \]
We write down 3 as another factor.

4. **Continue with the new quotient (15)**:
Now, we take 15 and check if it is divisible by 3.
\[ 15 \div 3 = 5 \]
We write down another 3 as a factor.

5. **Continue with the new quotient (5)**:
Finally, we take 5. Since 5 is a prime number, we divide it by itself.
\[ 5 \div 5 = 1 \]
We write down 5 as a factor.

6. **Compile the prime factors**:
Now we have all the prime factors. The prime factorization of 90 can be expressed as:
\[ 90 = 2 \times 3 \times 3 \times 5 \]
This can also be written using exponents as:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

### Final Answer:
The prime factorization of 90 is \( 2^1 \times 3^2 \times 5^1 \).