Question

The largest number among the following that will perfectly divide 101100 - 1 is?

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

The largest number among the following that will perfectly divide $\sf{( {101}^{100} - 1) \: \: \: is }$

a) 100

b) 1000

c) 10000

d) 100000

EVALUATION

Here the given given expression is

$\sf{( {101}^{100} - 1) }$

We now expand it as below

$\sf{( {101}^{100} - 1) }$

$\sf{ = {(1 + 100)}^{100} - 1 }$

$\sf{ = { \bigg(1 + {}^{100}C_1. {100}^{} +{}^{100}C_2. {100}^{2} + {}^{100}C_3. {100}^{3} + .. + {}^{100}C_{100}. {100}^{100} \bigg) - 1 } }$

$\sf{ = { \bigg(1 + 100. {100}^{} +{}^{100}C_2. {100}^{2} + {}^{100}C_3. {100}^{3} + .. + {}^{100}C_{100}. {100}^{100} \bigg) - 1 } }$

$\sf{ = { \bigg(1 + {100}^{2} +{}^{100}C_2. {100}^{2} + {}^{100}C_3. {100}^{3} + .. + {}^{100}C_{100}. {100}^{100} \bigg) - 1 } }$

$\sf{ = { \bigg( {100}^{2} +{}^{100}C_2. {100}^{2} + {}^{100}C_3. {100}^{3} + .. + {}^{100}C_{100}. {100}^{100} \bigg) } }$

$\sf{ = {100}^{2} { \bigg( 1 +{}^{100}C_2 + {}^{100}C_3. {100}^{} + .. + {}^{100}C_{100}. {100}^{98} \bigg) } }$

$\sf{ = 10000 { \bigg( 1 +{}^{100}C_2 + {}^{100}C_3. {100}^{} + .. + {}^{100}C_{100}. {100}^{98} \bigg) } }$

Hence the largest number among the following that will perfectly divide $\sf{( {101}^{100} - 1) \: \: \: is }$ 10000

FINAL ANSWER

Hence the correct option is c) 10000

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