Give the prime factorization of each of the following numbers :
`1224`

Options

A. `2^(3) xx 3^(2) xx 17`
B. `2^(3) xx 3^(2) xx 19`
C. `2^(3) xx 3^(2) xx 15`
D. None of these

Answer

To find the prime factorization of the number 1224, we will divide the number by the smallest prime numbers until we reach 1. Here’s how we can do it step by step:

### Step 1: Start with the number 1224
We will begin by dividing 1224 by the smallest prime number, which is 2.

### Step 2: Divide by 2
1224 ÷ 2 = 612

### Step 3: Divide by 2 again
Now, we take the result (612) and divide it by 2 again.
612 ÷ 2 = 306

### Step 4: Divide by 2 once more
Next, we divide 306 by 2 again.
306 ÷ 2 = 153

### Step 5: Now divide by 3
Since 153 is not divisible by 2, we move to the next prime number, which is 3.
153 ÷ 3 = 51

### Step 6: Divide by 3 again
Now, we divide 51 by 3.
51 ÷ 3 = 17

### Step 7: Finally, divide by 17
Now, we have 17, which is a prime number itself.
17 ÷ 17 = 1

### Step 8: Collect the prime factors
Now we can collect all the prime factors we used in our divisions:
- We divided by 2 three times (2 x 2 x 2)
- We divided by 3 two times (3 x 3)
- We divided by 17 one time (17)

### Step 9: Write the prime factorization
Thus, the prime factorization of 1224 can be written as:
\[ 1224 = 2^3 \times 3^2 \times 17 \]