68 questions linked to/from Highest power of a prime $p$ dividing $N!$
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### How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
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### Prime powers that divide a factorial [duplicate]

If we have some prime $p$ and a natural number $k$, is there a formula for the largest natural number $n_k$ such that $p^{n_k} | k!$. This came up while doing an unrelated homework problem, but it is ...
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### Exponent of Prime in a Factorial [duplicate]

I was just trying to work out the exponent for $7$ in the number $343!$. I think the right technique is $$\frac{343}{7}+\frac{343}{7^2}+\frac{343}{7^3}=57.$$ If this is right, can the technique be ...
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### What is the largest power of 2 that divides $200!/100!$. [duplicate]

What is the largest power of 2 that divides $200!/100!$. No use of calculator is allowed. I had proceeded in a brute force method which i know regret.. I would like to know your methods.
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### How many factors of $10$ in $100!$ [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How many factors of 10 are there in $100!$ (IIT Question)? Is it 26,25,24 or any other value Please tell how you have done it
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### the exponent of the highest power of p dividing n! [duplicate]

The formula for the exponent of the highest power of prime $p$ dividing $n!$ is $\sum \frac{n}{p^k}$, but the question is $n=1000!$ (really, it has the factorial) and $p=5$. When I use Wolfram Alpha ,...
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### The method of solving for a factor of $90!$ [duplicate]

If $90! = (90)(89)(88)...(2)(1)$, then what is the exponent of the highest power of $2$ which will divide $90!$ ? How would I apply one of the easiest method from Here? I need help on applying the ...
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### Concecutive last zeroes in expansion of $100!$ [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ In decimal form, the number $100!$ ends in how many consecutive zeroes. I am thinking of the factorization of $100!$ but I am stuck. ...
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### Determine the number of 0 digits at the end of 100! [duplicate]

I got this question, and I'm totally lost as to how I solve it! Any help is appreciated :) When 100! is written out in full, it equals 100! = 9332621...000000. Without using a calculator, determine ...
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### Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number. [duplicate]

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number. Actually, I know a way to solve this, but even if it is very large and ...
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### Dividing $61!$ by $3$ as we can. [duplicate]

Find maximum possible $n$ in the equation $61!=3^n\cdot m$. Some textbooks gives a solution to this question like this: \begin{align*} 61&=\textbf{20}\cdot 3+1 \\ 20&=\textbf{6}\cdot 3+2 \\ 6&...
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### Why does $\sum\limits_{k=1}^\infty \lfloor m/(n^k)\rfloor$ give you the number of times that $n$ divides $m!$? [duplicate]

If $n$ is a prime less than $m$, with $n,m \in \mathbb N$, why does $$\sum_{k=1}^\infty \left\lfloor \frac{m}{n^k}\right\rfloor$$ give you the number of times that $n$ divides $m!$? Examples: $n=13$ ...
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### What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + \left\lfloor\frac{...
### Prove the multiplicity property for $n!$ [duplicate]
I was given this hint in a different problem, Now use that a prime $p$ occurs in $n!$ with multiplicity exactly \$\lfloor n/p\rfloor + \lfloor n/p^2\rfloor + \lfloor n/p^3\rfloor + \lfloor n/p^4\...