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Find the number of zeros at the end of $15! + 16! + 17! + 18!$ ? I know the method find the number of zeros at the end of x! where $x = { 15! , 16! , 17! ...}$ by dividing by number by $5,5^2, 5^3$ and so on .

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  • $\begingroup$ This will be that except that you're calculating $15!(1+16+16\cdot17+16\cdot17\cdot18)$, $\endgroup$
    – user694818
    Commented Sep 8, 2019 at 12:02
  • $\begingroup$ Wolframalpha tells me it ends in $4$ zeroes. $\endgroup$ Commented Sep 8, 2019 at 12:08
  • $\begingroup$ yes ,It ends in 4 zeros .But can you tell me any optimized way of finding it $\endgroup$ Commented Sep 8, 2019 at 12:10

2 Answers 2

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$1+16+16 \cdot 17+ 16 \cdot 17 \cdot 18 = 5185$. If the first non-zero digit of $15!$ is odd, then there will be just $3$ zeroes at the end from the $15!$. However, if it is even, there will be exactly one more zero. (hint: look at integers from $1$ to $9$)

Then we have to find:

$$1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 11 \cdot 12 \cdot 13 \cdot 14 \pmod {10}$$

and the result is an even number.

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  • $\begingroup$ You mean to have $\pmod{1000}$? $\endgroup$ Commented Sep 8, 2019 at 13:34
  • $\begingroup$ No, just $\pmod {10}$. This is just the same as the last non-zero digit of $15! \pmod {10 \ 000}$. $\endgroup$
    – Toby Mak
    Commented Sep 8, 2019 at 13:36
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    $\begingroup$ Ah my bad :p ${}$ $\endgroup$ Commented Sep 8, 2019 at 13:37
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First divide $15!$ by $10^3$ (in the obvious way, from its expression) to determine its last nonzero digit. Hopefully, it will be the 4th digit from the right. You're left with $$N=3\cdot 4\cdot 6\cdot 7\cdot 8\cdot 9\cdot 11\cdot 12\cdot 13\cdot 7\cdot 3= 2^8\cdot 3^6\cdot 7^2\cdot 11\cdot 13$$ Modulo $100$, we have $$N\equiv 56\cdot 29\cdot 49\cdot 43=56\cdot 29\cdot (50-1)(40+3)\equiv 56\cdot 29\cdot 7\equiv 56\cdot 3\equiv 68 $$ As a conclusion, $\; 15!\equiv 68\,000\mod 10^5$.

Now we'll compute: \begin{align}M&=1+16+16\cdot 17+16\cdot 17\cdot 18=17^2+16\cdot 17\cdot 18=17(17+17^2-1)\\ &= 17(17+289-1)=17\cdot 305=5185 \end{align}

Thus, $$15!+16!+17!+18!\equiv68\,000\cdot 5185\equiv 8000\cdot 85=680\,000\equiv \color{red}{ 80000} \mod 10^5.$$

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