Sorry, not even close.
is a factorization of 288, BUT...
6 is not prime, therefore any factorization that contains a 6 is NOT a prime factorization.
Here's the process:
Start with the given number
288. This number is even so it is divisible by 2.
144. This number is even so it is divisible by 2.
72. This number is even so it is divisible by 2.
36. This number is even so it is divisible by 2.
18. This number is even so it is divisible by 2.
9. This number is NOT even so it is NOT divisible by 2. This number is divisible by 3.
3 is prime. Two factors of 3.
All together, 5 factors of 2 and 2 factors of 3
is the unique prime factorization of 288.
The illustrated process works for finding the prime factorization of any integer. Always start with 2. Numbers are divisible by 2 if they are even. Once you arrive at an odd result, move on to 3. If the sum of the digits is divisible by 3, the number is divisible by 3. Once you get a result that is not divisible by 3, go on to the next higher prime number, namely 5. Numbers that end in either 5 or 0 are divisible by 5. 7 doesn't have a convenient rule -- trial and error works well though. If the center digit of a three digit number is the sum of the first and last digits, the number is divisible by 11, but you need to do trial and error anyway. Continue in this pattern taking the next higher prime number as your divisor until you either clearly achieve a prime result or the next trial divisor (the next higher prime) is greater than the square root of the dividend.
For example, the prime factorization of 947.
Square root is a little bigger than 30.
Not even, so no factors of 2.
Sum of digits is 20, 20 not divisible by 3, so no factors of 3.
Does not end in 0 or 5, so no factors of 5.
Calculator has a decimal fraction result when divide by 7, so no factors of 7.
Same thing when dividing by 11, 13, 17, 19, 23, 29.
The next prime number is 31 which is greater than the square root of 947
Conclusion: 947 is prime.
John