Page 52 - Numbertheory

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.( ) ()

n = 3k +1 n = 3k

. n 2 = 3(3k 2) n = 3k . n = 3k + 2

() . n 2 = 3M +1 n = 3k + 2 n = 3k +1

. ()

. n = 2k +1 n = 2k

n = 2k +1 . n 2 = 2(2k 2) n = 2k

. () ( ) n2 = 4M +1

[AHSME 1951] ( )

n n3 −n

6( ) 4( ) 3( ) 2 ()

:( )

n 3 − n = (n −1)n (n +1)

.

2

. lcm (2, 3) = 6 n3 −n 3

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