Question

Find out the remainder when 78 factorial is divided by 83? Pls give the solution ​

Answer 1

Given:

78 factorial is divided by 83

To find:

Find out the remainder when 78 factorial is divided by 83?

Solution:

Using Wilson theorem, we have,

if n is a prime number, then (n-1)! + 1 is divisible by n.

⇒ (n-1)! ≡ -1 (mod n) ....(1)

since, 83 is a prime number, using (1) we get,

82! ≡ -1 (mod 83)

82 × 81 × 80 × 79 × 78! ≡ -1 (mod 83)

(83 - 1) × (83 - 2) × (83 - 3) × (83 - 4) × 78! ≡ -1 (mod 83)

(- 1) × ( - 2) × ( - 3) × ( - 4) × 78! ≡ -1 (mod 83)

24 × 78! ≡ -1 (mod 83)

24 × 78! ≡ 82 (mod 83)

12 × 78! ≡ 41 (mod 83)

12 × 78! ≡ - 42 (mod 83)

2 × 78! ≡ - 7 (mod 83)

2 × 78! ≡ 76 (mod 83)

78! ≡ 38 (mod 83)

∴ 38 is the remainder when 78! is divided by 83.