证明：70!,61!对模71同余

70!-61!
=70*69*68*67*66*65*64*63*62*61!-61!
=(70*69*68*67*66*65*64*63*62-1) *61!
因为 70*69*68*67*66*65*64*63*62 mod 71
=(71-1)(71-2)(71-3)(71-4)(71-5)(71-6)(71-7)(71-8)(71-9) mod 71
= -9!mod 71
= -7!(8*9) mod 71
= -7!(71+1) mod 71
= -7!mod 71
= -5040 mod 71
= -(71*71)+1 mod 71
=1 mod 71
所以
70*69*68*67*66*65*64*63*62-1 mod 71=0
所以 70!-61!mod 71=0
命题得证

190+9942.792
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