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           .     .12                      3 x()                 )
                                             x ()               )
         n    . n3
              6( )                       n [AMC12B 2010] (
    5( )
                           n                n2  20
53
53                                        7( )    8 ()
                                                       (
                                         n
                                                  . 12

n + 60 ( )    n +12 ( )       n+5 ( )                 12

                                                n+3 ()          )
                                                n(

              gcd(n, 2n +1) = 1 gcd(2n, 3n ) = n ( )

              gcd(n, 2n +1) = n gcd(2n, 3n) = 1 ( )

              gcd(2n,3n) = gcd(n, 2n +1) = 1 ( )

              gcd(2n, 3n) = gcd(n, 2n +1) = n ( )

            . S = 1!+ 2!+ 3!+ ... + 99!         [AHSME 1978] (  )

                                                S

3() 5( ) 8( )                                      9 ()

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