Find a positive integer x such that the last three digits of

Find a positive integer x such that the last three digits of 7^7x are 007.

Solution

Given that the last three digits of 7^7x is 007

so let x=1

=> 7⁷⁽¹⁾ = 823543

the last three digits are not 007

   let x=2

=> 7⁷⁽²⁾ = 7¹⁴

     =678223072849

the last three digits are not 007

   let x=3

=> 7⁷⁽³⁾

=7²¹

  =558545864083284007

the last three digits are 007

so x=3