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# Number Theory

0
70
2

If I expand 25*24*23* ... *12*11*10, how many zeros are there at the end of the number I get?

Aug 5, 2021

#1
+76
-1

Since it's obvious there's more 2's than 5's in this string of numbers, we just count the number of 5's to get the answer/

5's: 10,15,20,25(25has 5^2), so there is 5^5 in total

from that we know there's 100000 that can be combined in this number which leaves us with 5 0's at the end

Aug 5, 2021
#2
+26213
+3

If I expand
$$25*24*23* \dotsm *12*11*10$$,
how many zeros are there at the end of the number I get?

$$25*24*23* \dotsm *12*11*10 = 25!-9!$$

$$\begin{array}{|rcll|} \hline \text{zeros} &=& \lfloor\frac{25}{5}\rfloor +\lfloor\frac{25}{25}\rfloor -\lfloor\frac{9}{5}\rfloor \\ &=& 5+1-1 \\ \mathbf{\text{zeros}} &=& \mathbf{5} \\ \hline \end{array}$$

Aug 5, 2021