Question

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

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

Answer 1

Answer: (A), (B), (C)

The correct options are
A 100
B 1000
C 10000

$(1+100{)}^{100}-1$
$=1+100\left(100\right)+{}^{100}{C}_2{100}^2+{}^{100}{C}_3{100}^3+...{}^{100}{C}_{100}{100}^{100}-1$
$=100\left(100\right)+{}^{100}{C}_2{100}^2+{}^{100}{C}_3{100}^3+...{}^{100}{C}_{100}{100}^{100}$
$={10}^4.k$
Where
$k$ is positive integer divisible by 10,100,1000 and 10,000.