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What is the number of zeros in the decimal expansion of $11^{100}-1$?

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    According to python 12. Your number is 137806123398222701841183371720896367762643312000384664331464775521549852095523076769401159497458526446000.2012-10-25
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    how can it be solved using some theorems?2012-10-25
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    Or if that's binary, you have 3^4 - 1 = 80, so the answer is 1 zero in the decimal expansion.2012-10-25
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    You can easily show that $11^{100} - 1$ has exactly $3$ trailing $0$'s. However, the $0$'s in the middle of the number are not really very amenable to analysis.2012-10-25

2 Answers 2

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11^100 - 1 = 137806123398222701841183371720896367762643312000384664331464775521549852095523076769401159497458526446000 which has 12 zeros.

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We ca see that,

$$11^{100}-1=(10+1)^{100}-1=(10^{100}+100\cdot10^{99}+99\cdot50\cdot 10^{98}+...+1)-1$$

Now, try to investigate the above.