Elementary theory of numbers
Number Theory
Level
pending
number [{(3!)!}!] written in decimal system has more than thousand digits, find the number of zeros at the end of the expansion.
The answer is 178.
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1 solution
We have 3! = 6,3!! = 6! = 720,3!!! = 720! > 99! 100621 > 101242. Thus, the number 3!!! has more than thousand digits. By the well-know theorem where [xl denotes the greatest integer ≤ x. It follows that the largest power of 5 which divides 3!!! = 720! is [ 720/5]+[720/25]+[ 720/125+[720/125] = 144+28+5+1 = 178 ' while the largest power of 2 dividing 720! is still greater (since already [ 720/2] = 360). It follows that the number 3!!! has 178 zeros at the end of its decimal expansion