The greatest integer which divides the number 101^ 100 - 1, is - Vidyadip

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) ${(1 + 100)^{100}} = 1 + 100.100 + \frac{{100.99}}{{1.2}}.{(100)^2} + \frac{{100.99.98}}{{1.2.3}}{(100)^3} + ....$

${(101)^{100}} - 1 = 100.100\left[ {1 + \frac{{100.99}}{{1.2}} + \frac{{100.99.98}}{{1.2.3}}.100 + ....} \right]$

From above it is clear that,

${(101)^{100}} - 1$ is divisible by $(100)^2$ $= 10000$

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