The trend of huge number (2)
2 0 1 6 ! 2 0 1 7 ! 2 0 1 8 ! × 2 0 1 5 ! 2 0 1 6 ! 2 0 1 7 ! ≡ ? ( m o d 2 0 1 7 )
Notation : ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 .
Hints :You may find Wilson's theorem useful.
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1 solution
In general, we have ( ( p − 1 ) ! ) m × ( ( p − 2 ) ! ) n ≡ ( − 1 ) m ( m o d p ) for all odd primes p and non-negative integers m , n . This follows from Wilson's Theorem and it's extension that ( p − 2 ) ! ≡ 1 ( m o d p ) for all odd primes p .
2 0 1 7 is a prime
( 2 0 1 7 − 1 ) ! ≡ − 1 ( m o d 2 0 1 7 )
2 0 1 6 ! ≡ − 1 ( m o d 2 0 1 7 )
2 0 1 6 ! 2 0 1 7 ! ≡ ( − 1 ) 2 0 1 7 ! ( m o d 2 0 1 7 )
2 0 1 6 ! 2 0 1 7 ! ≡ ( − 1 ) 2 n ( m o d 2 0 1 7 ) , where n is a positive integer.
2 0 1 6 ! 2 0 1 7 ! ≡ 1 ( m o d 2 0 1 7 )
2 0 1 6 ! 2 0 1 7 ! 2 0 1 8 ! ≡ 1 ( m o d 2 0 1 7 )
2 0 1 6 ! ≡ − 1 ( m o d 2 0 1 7 )
2 0 1 6 ! ≡ 2 0 1 6 ( m o d 2 0 1 7 )
2 0 1 6 2 0 1 6 ! ≡ 2 0 1 6 2 0 1 6 ( m o d 2 0 1 7 )
2 0 1 5 ! ≡ 1 ( m o d 2 0 1 7 )
2 0 1 5 ! 2 0 1 6 ! 2 0 1 7 ! ≡ 1 ( m o d 2 0 1 7 )
2 0 1 6 ! 2 0 1 7 ! 2 0 1 8 ! × 2 0 1 5 ! 2 0 1 6 ! 2 0 1 7 ! ≡ 1 × 1 ( m o d 2 0 1 7 )
2 0 1 6 ! 2 0 1 7 ! 2 0 1 8 ! × 2 0 1 5 ! 2 0 1 6 ! 2 0 1 7 ! ≡ 1 ( m o d 2 0 1 7 )