# Find the Last Five Digits

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• May 27th 2012, 01:14 PM
aakgoel
Find the Last Five Digits
Please help me do this question.

Find the last five digits of

(101^100) -1
• May 28th 2012, 01:26 AM
BobP
Re: Find the Last Five Digits
10000

Use the binomial expansion for $(1+100)^{100}.$
• May 30th 2012, 06:41 AM
aakgoel
Re: Find the Last Five Digits
Dear Sir thank you for your consideration.
But how would the binomial expansion of the same help as then I would have to add all the numbers
and then get my results.

Please help and give a proper explanation if possible.

Regards
aakgoel
• May 30th 2012, 06:57 AM
emakarov
Re: Find the Last Five Digits
$(1 + 100)^{100} - 1 = \binom{100}{1}100+\binom{100}{2}100^2+\binom{100}{ 3}100^3+\dots$. All terms starting from the third one end with at least 6 zeros, so you need to consider the first two terms only.