Large factorial number divisibility

Question

How can I proof that 14! is not divisible by 3⁶?

Answer 1

There are $4=\lfloor 14/3\rfloor$ multiples of $3$ and $1=\lfloor 14/9\rfloor$ multiple of $9$.

Hence $3^5$ divides $14!$ but $3^6$ does not.

Answer 2

Count the factors of $3$ that appear. You get one from each multiple of $3$ except two from multiples of $9$ (and, if your factorial were larger, three from multiples of $27$, etc.) How many do you get? Another approach is to ask Alpha and get $14!=87178291200 = 2^{11}×3^5×5^2×7^2×11×13 $

Answer 3

We have $$ 14! = 1\cdot2\cdot3\cdot4\cdot5\cdot6\cdots13\cdot14\\ = 2\cdot3\cdot2^2\cdot5\cdot(2\cdot3)\cdots13\cdot(2\cdot7)\\ = 2^{11}\cdot3^{?}\cdot5^2\cdot7^2\cdot11\cdot13 $$ Just count the number of $3$'s you get in line two above.