The number $101^{100} - 1$ is divisible by

100

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1000

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10000

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100000

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Solution

The correct options are
A 100
B 1000
C 10000
$(1 + 100)^{100} - 1$
$= 1 + 100 (100) +^{100} C_{2} 100^{2} +^{100} C_{3} 100^{3} + . . .^{100} C_{100} 100^{100} - 1$
$= 100 (100) +^{100} C_{2} 100^{2} +^{100} C_{3} 100^{3} + . . .^{100} C_{100} 100^{100}$
$= 10^{4} . k$
Where
$k$ is positive integer divisible by 10,100,1000 and 10,000.

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