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Question

Mathematics Question on binomial expansion formula

The number (101)100−1(101)^{100} - 1 is divisible by

A

10410^4

B

10610^6

C

10810^8

D

101210^{12}

Answer

10410^4

Explanation

Solution

Now, (101)100−1=(1+100)100−1(101)^{100}-1=(1+100)^{100}-1
=100C1100+100C21002+…+100C100100100={ }^{100} C_{1} 100+{ }^{100} C_{2} 100^{2}+\ldots+{ }^{100} C_{100} 100^{100}
=(100)2[1+100C2+100C31000+…]=(100)^{2}\left[1+{ }^{100} C_{2}+{ }^{100} C_{3} 1000+\ldots\right]
Hence, it is divisible by 10410^{4}.