



 
Brandon continued.
Hi Brandon, I want to look at a similar but smaller question.
Analogous to what you did I wrote 180 in terms if its prime factors, $180 = 2^2 \times 3^2 \times 5$ and 180 divided by 5 is 36. You can list the multiples of 5 that divide 181!, they are
Dividing each of these by 5 gives
Now you can see the problem, every time a multiple of $5 ^2 = 25$ divides 180 you are left with an extra 5. $25 \times 7 = 175$ so there are 7 multiples of 25 that divide 180! Hence 5 divides 180! at least $36 + 7 = 43$ times. What about $5^{3} = 175?$ 175 divides 180 once so there is one more 5 that divides 181! and hence $5^{44}$ divides 181!. Thus 44 is that largest positive integer for which $5^{k}$ divides 181! What remains is to show that $(2^2)^{44}$ and $(3^2)^{44}$ divide 181! Penny 



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