These Numbers Are Way Too Huge!
Find the greatest common divisor of the two numbers and .
Note:
2017 is
prime
.
Notation:
is the
factorial
notation. For example,
.
The answer is 2017.
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1 solution
According to Wilson's theorem ,
( p − 1 ) ! ≡ − 1 ( m o d p )
for prime p. Since 2017 is a prime, hence ( 2 0 1 6 ) ! + 1 ≡ 0 ( m o d 2 0 1 7 ) ⇒ 2 0 1 6 ! + 1 i s a m u l t i p l e o f 2 0 1 7 .
For all prime p < 2 0 1 7 , ( 2 0 1 6 ) ! ≡ 0 ( m o d p ) ⇒ ( 2 0 1 6 ) ! + 1 ≡ 1 ( m o d p )
hence 2016!+1 is not a multiple of any prime number smaller than 2017.
The greatest prime divisor of 2017! is 2017. Since this is the only prime divisor shared by these two numbers, it is the greatest common divisor.