MCQ
The greatest integer which divides the number ${101^{100}} - 1$, is
- A$100$
- B$1000$
- ✓$10000$
- D$100000$
✓
Answer
Correct option: C.
$10000$
c
(c) ${(1 + 100)^{100}} = 1 + 100.100 + \frac{{100.99}}{{1.2}}.{(100)^2} + \frac{{100.99.98}}{{1.2.3}}{(100)^3} + ....$
(c) ${(1 + 100)^{100}} = 1 + 100.100 + \frac{{100.99}}{{1.2}}.{(100)^2} + \frac{{100.99.98}}{{1.2.3}}{(100)^3} + ....$
${(101)^{100}} - 1 = 100.100\left[ {1 + \frac{{100.99}}{{1.2}} + \frac{{100.99.98}}{{1.2.3}}.100 + ....} \right]$
From above it is clear that,
${(101)^{100}} - 1$ is divisible by $(100)^2$ $= 10000$
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