Is this divisible by 11 ?

99887766554433221 12233445566778899 99887766554433221 - 12233445566778899

Is the above number divisible by 11 11 ?

Yes No

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2 solutions

Yuriy Kazakov
Aug 13, 2020

( 99887766554433220 + 1 ( 10000000000000000 + 2233445566778899 ) m o d ( 11 ) = (99887766554433220+1-(10000000000000000 +2233445566778899) \mod(11) =

1 ( 10000000000000000 ) m o d ( 11 ) = 9999999999999999 m o d ( 11 ) = 0 1-(10000000000000000) \mod(11)= -9999 9999 9999 9999 \mod(11)=0

Let the given number be N N . Then

N 99887766554433221 12233445566778899 (mod 11) 998877665544332 3 1 10 1 3 233445566778899 + 1 0 15 (mod 11) 1 0 15 10 (mod 11) 99999999999999 Number of 9s = 14 0 + 10 10 (mod 11) 99999999999999 Number of 9s = 14 0 (mod 11) 0 (mod 11) \begin{aligned} N & \equiv 99887766554433221-12233445566778899 \text{ (mod 11)} \\ & \equiv 998877665544332\blue31 - \blue{10} -1\red3233445566778899 + \red{10^{15}} \text{ (mod 11)} \\ & \equiv \blue{10^{15}} - 10 \text{ (mod 11)} \\ & \equiv \blue{\underbrace{99999999999999}_{\text{Number of 9s}=14}0 + 10} - 10 \text{ (mod 11)} \\ & \equiv \underbrace{99999999999999}_{\text{Number of 9s}=14}0 \text{ (mod 11)} \\ & \equiv \boxed 0 \text{ (mod 11)} \end{aligned}

Yes , N N is divisible by 11 11

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