Find It Without Plugging Values
n ! + 1 ( n + 1 ) ! + 1
If n is a positive integer , find the greatest common divisor of the two numbers above.
Notation : ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 .
The answer is 1.
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1 solution
Nice problem! What happens if we replace the + with a -?
I don’t rlly understand that. Could you describe it in a more simple way
Another way is to use the Euclidean Algorithm: g c d ( n ! + 1 , ( n + 1 ) ! + 1 ) = g c d ( n ! + 1 , ( n + 1 ) ! + 1 − ( n ! + 1 ) ) = g c d ( n ! + 1 , ( n + 1 ) ! − n ! ) = g c d ( n ! + 1 , n ! n ) . Now, for any p > 1 such that p ∣ n ! n , we have that p ∣ n ! , and so it does not divide n ! + 1 . It only divides n ! + 1 when p is essentially 1 .
Let p be a prime factor of n ! + 1 , then p > n , because p ∤ n ! . Note that ( n + 1 ) ! + 1 = ( n + 1 ) n ! + 1 = n ⋅ n ! + n ! + 1 , as p ∣ ( n ! + 1 ) , then p ∤ ( ( n + 1 ) ! + 1 ) , cause p ∤ n ⋅ n ! . Therefore, g c d ( n ! + 1 , ( n + 1 ) ! + 1 ) = 1