The number (101)100 - 1 is divisible by

Question

The number (101)100 - 1 is divisible by (A) 104 (B) 106 (C) 108 (D) 1012

Tardigrade Admin · Accepted Answer

(101)100-1 =(1+100)100-1 = (1+ 100 C1 ⋅ 100+ 100 C2 1002+ ldots ldots)-1 = 100 C1 100+ 100 C2(100)2+ 100 C3(100)3+ ldots ldots .+ 100 C100(100)100 = 104(1+ 100 C2+ 100 C3 102+ ldots. + 100 C100(100)98 = 104(1+ an integer multiple of 10)

Q. The number $(101)^{100} - 1$ is divisible by

$1 0^{4}$

$1 0^{6}$

$1 0^{8}$

$1 0^{12}$

Q. The number (101)100−1 is divisible by

104

106

108

1012

(101)100−1=(1+100)100−1 =(1+100C1​⋅100+100C2​1002+……)−1 =100C1​100+100C2​(100)2+ 100C3​(100)3+…….+100C100​(100)100 =104(1+100C2​+100C3​102+… +100C100​(100)98 =104(1+ an integer multiple of 10)

Q. The number $(101)^{100} - 1$ is divisible by

$1 0^{4}$

$1 0^{6}$

$1 0^{8}$

$1 0^{12}$