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How many zeros are at the end of (100!)(200!)(300!)(400!) when multiplied out?

 May 1, 2022
 #1
avatar+168 
+1

(100/5 + 100/25 + 200/5 + 200/25 + 200/125 +300/5 + 300/25 + 300/125 + 400/5 + 400/5 + 400/5)= 387 zeroes.

 May 1, 2022
 #3
avatar+168 
+1

oops I made a typo on this when calculating, ignore. 

HoldMyBeer  May 1, 2022
 #2
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+1

Floor{[100/5]+[100/25]+[200/5]+[200/25]+[200/125]+[300/5]+[300/25]+[300/125]+[400/5]+[400/25]+[400/125]}==246 zeros:

 

144000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000==246 zeros.

 May 1, 2022
 #4
avatar+9459 
+1

We calculate the trailing zeroes of each of 100!, 200!, 300!, and 400!.

 

Note that we have this nice formula for calculating the trailing zeroes of n! : \(\#\text{ trailing zeros} = \displaystyle\sum_{k = 1}^\infty \left\lfloor\dfrac{n}{5^k}\right\rfloor\).

Then \(100!\) has \(\left\lfloor\dfrac{100}5\right\rfloor + \left\lfloor\dfrac{100}{5^2}\right\rfloor = 24\) trailing zeros.

 

Similarly, 200! has 49 trailing zeros, 300! has 74 trailing zeros, and 400! has 99 trailing zeros.

 

Then, (100!)(200!)(300!)(400!) has 24 + 49 + 74 + 99 = 246 trailing zeros.

 May 2, 2022

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