vinay89 wrote:so number of 23 in 300! will determine the number of 24! in 300!
Dont really get why we look for the greatest prime factor?
300 will contain way more of the smaller prime factors. Thus, the number of times the greatest prime factor (of 24) can go into 300 will limit the number of times 24! can go into 300!
For example, if we think about the really small prime factors of 24--the 2s and the 3s, it is clear that there are way more of them in 300. Let's think about the second biggest prime factor of 24--19. Every 19th number is a multiple of 19. But every 23rd number is a multiple of 23. So, there are fewer 23s in 300 than there are 19s. (300/19 = 15.7. Thus, there are 15 19s in 300. This doesn't mean that 24! can divide 300! 15 times, however, because there are only 13 23s in 24!. This means that 24! can't divide 300 a 14th time).
