Dubious Selection
( 3 9 3 9 7 8 7 8 ) m o d 7 8 7 9 = ?
Note : You may use the fact that 7879 is prime.
The answer is 7878.
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2 solutions
Nice! I knew Wilson's Theorem was the key, (which is why I just guessed at 7 8 7 8 as the answer), but I didn't know about the refinement, (which finally made sense to me after a few minutes). Thanks for this new bit of knowledge. :)
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I answered − 1 , which is technically correct but not recognized by the website :)
Anyway, here is a more elementary solution: Note that 7 8 7 8 ≡ − 1 , 7 8 7 7 ≡ − 2 , … , 3 9 4 0 ≡ − 3 9 3 9 modulo 7879. Therefore ( 7 8 7 8 3 9 3 9 ) ≡ 3 9 3 9 ⋅ 3 9 3 8 ⋯ 1 3 9 4 0 ⋅ 3 9 4 1 ⋯ 7 8 7 8 ≡ 3 9 3 9 ⋅ 3 9 3 8 ⋯ 1 ( − 3 9 3 9 ) ⋅ ( − 3 9 3 8 ) ⋯ ( − 1 ) ≡ 3939 times ( − 1 ) ⋅ ( − 1 ) ⋯ ( − 1 ) ≡ ( − 1 ) 3 9 3 9 ≡ − 1 . (It is essential that the base of modulo, 7879, is prime. Otherwise we would end up dividing by a zero divisor, which is undefined.)
Did it the same way
Observed features for prime numbers from 3 to 53 and realized. 7878 C 3939 must be an integer!
If not 7878, then it must be 1. Tried with 7878 correct! Calculator cannot tell the answer; it only tells a wrong answer of // 6526.4351304442254481560126764616 //.
For X: Not prime and V: Prime for the following,
...
7875 X
7877 V
7879 V
7881 X
7883 V
7885 X
...
Most likeliness for 7879 Prime is having a remainder of 7878.
Answer: 7878
Wilson's Theorem does the job. With p = 7 8 7 9 and m = 2 p − 1 = 3 9 3 9 , we know that ( p − 1 ) ! ≡ p − 1 ( m o d p ) and ( m ! ) 2 ≡ ( − 1 ) m + 1 = 1 ( m o d p ) , by a straightforward refinement of Wilson's Theorem. Thus ( 3 9 3 9 7 8 7 8 ) = ( 3 9 3 9 ! ) 2 7 8 7 8 ! ≡ 7 8 7 8 ( m o d 7 8 7 9 )
The refinement of Wilson's Theorem is based on the factorization − 1 ≡ ( p − 1 ) ! = 1 ∗ 2 ∗ . . . ∗ m ∗ ( p − m ) ∗ ( p − ( m − 1 ) ) ∗ . . . ∗ ( p − 1 ) ≡ ( m ! ) 2 ( − 1 ) m ( m o d p )