Question

The greatest integer which divides the number $101^{100} - 1$ is

• A. 100
• B. 1000
• C. 10000
• D. 100000

Answer 1

Answer: (C)

10000

Answer 2

$(1 + 100)^{100} = 1 + 100.100 + \frac{100.99}{1.2}(100)^{2} + ....$

⇒$101^{100} - 1 = 100.100\left\lbrack 1 + \frac{100.99}{1.2} + \frac{100.99.98}{3.2.1}100 + ..... \right\rbrack$

From above it is clear that, $101^{100} - 1$ is divisible by $(100)^{2} = 10000$