Factor the number into primes only -- primes are numbers whose only factors are 1 and themselves. So to factor 50, it is not enough to note that 50 = 2 x 25. That's because 25 is composite, so it can be factored further. Keep factoring until all the factors are prime: 50 = 2 x 5 x 5. Note that if the number you are factoring is prime, the factoring process will be very simple. For example, to factor 107, you can only say 1 x 107 = 107.
Arrange the primes that appear in the factorization, noting the duplications. You should note any primes that are missing in the factorization up to the largest prime that is represented. For example, the first few primes in order are 2, 3, 5, 7, 11 and 13. To factor 100, you first find the prime factors: 100 = 2 x 2 x 5 x 5. Arrange the primes in ascending order and note the duplications, so the factorization of 100 has these primes: 2 (2 of these), 3 (none of these), 5 (two of these), 7 or higher (none of these). In other words, 100 could be described as 2 0 2 -- and no other number could be described with this unique string of exponents.
Notice how the unique representations of large composite numbers can be easily computed from their factors without going all the way to primes. For example, 100 has the unique factorization 2 0 2 and 24 = 2 x 2 x 2 x 3 has the unique factorization 3 1. You can add the exponents to get the unique factorization of 2,400 -- 2 0 2 + 3 1 = 5 1 2 -- as long as you remember to line up the prime exponential factors on the right instead of on the left, as is normal for addition.