My way of prime factorizing is to divide the number you are trying to find by the smallest prime factor. Take your result and divide again by the smallest prime factor of that result. Keep repeating this process until you get a prime number that can't be divided any further. The numbers you divided by and your final prime number are your prime factors. Remember that 1 is not a prime number!
It may help to see this using a factor tree (see image). 2 is the smallest prime number that divides 24, so break 24 down into 2 x 12. Now break down 12. Once again, 2 is the smallest prime number that divides 12. Break 12 down into 2 x 6. Now break down 6. Again, 2 is the smallest prime that divides 6, so break 6 down into 2 x 3. Since 3 is also prime, you've completely prime factorized 24. Your prime factors are circled in blue in the picture. Remember your final factors must all be prime - that's why it's called prime factorization!
Prime factorization of 24 = 2 x 2 x 2 x 3 = [tex]2^{3} \times 3 [/tex]