Wilson's Theorem Solved Example

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Wilson's Theorem:

In this video we will understand the application of Wilson's theorem to solve complex remainder problems with the help of an example.

Question: Find the remainder when 14! (read 14 factorial) is divided by 17.

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like you took -16 in place of 1, so are we always suppose to take negative remainder ? If not then how to decide when to tale +ve and when to take -ve remainder ?

shibasishroysarkar

what is the remainder when 17! is divided by 23?

himanshubhatt

What is mod p or any thing related to mod
I m nt getting it

arushimahajan

sir there is direct formula for reminder p -1/2 for this when p is prime

sourabhpatidar

when any no divided by D then remainder is 8 same no divided by 3D then remainder is 21 what is the remainder when twise of no divided by 3D .

shivamgupta-fiiq

what to do for question like ((22!/43))??

kashyapmakadia

i don't understand your solution and your accent

vanji

My Explanation :

We need to find 14! mod 17
According to wilson's theorem, (p-1) ! ≡ -1 (mod p), where p is a prime number.

By using wilson's theorem,
=> (17 - 1)! ≡ -1 (mod 17)
=> 16! ≡ -1 (mod 17)

Actually we need to find x such that x ≡ 14! (mod 17)
=> 15 * 16 * x ≡ 16! (mod 17)
=> 15 * 16 * x ≡ -1 (mod 17) (using wilson's theorem as mentioned above in (1))
=> (-2) * (-1) * x ≡ -1 (mod 17)
=> 2 * x ≡ -1 (mod 17)
=> x ≡ -1 * multiplicative_inverse(2) (mod 17) (for finding multiplicative inverse of 2, we need to find y such that 2*y (mod 17) = 1 => y = 9 satisfies)
=> x ≡ -1 * 9 (mod 17)
=> x ≡ -9 (mod 17)
=> x ≡ 8 (mod 17)

ismail

u can't understanding us properly

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