Well I don't know any short cut method to find prime factorization, but I guess you can look at the number in hand an at least figure out some of the prime factors(this is offcourse not possible always)

about the number 112,000

has got three tens.....

that is let 112,000 = n * 1000

now in 1000 how many odd prime factors you get?

3 odd prime factors, namely 5.... since 10 = 5 * 2.....

now in the residue, 112, this is obviously divisible by 2, well if you check for divisibility test of 3, it fails, obviously this is not divisible by 5(since 5/0 is not the last digit), try 7.... the next prime..... 7 * 16 = 112.....

now 16 is all having even prime factor namely 2....

therefore, the number of odd prime factor = three 5s + one 7 = 4 odd prime factor which is 1 less than number of total prime factors....