Number Theory or Algebra? Part 2 - Updated

Number Theory Level 3

Since the original Part 2 was the same problem posted by another member, I've come up with a new problem.

Find the remainder when $7917!$ is divided by $7919$

$7919$ is a prime number. The $1000^{th}$ one.

3 solutions

Sanchayapol Lewgasamsarn
Apr 28, 2014

Since that $7919$ is a prime number (you can write a C program to check this),

$7918! \equiv -1 (mod 7919)$

$7918 \cdot 7917! \equiv -1 (mod 7919)$

$7918 \cdot 7917! \equiv 7918 (mod 7919)$

$7917! \equiv 1 (mod \frac{7919}{(7918, 7919)})$

$7917! \equiv \boxed{1} (mod 7919)$

$\boxed{1}$ . FA, Final Answer.

i do not understand 1(mod7919). plz tell me

Arshad R Shaikh - 6 years, 10 months ago

By Wilson's Theorem.

Sanchayapol Lewgasamsarn - 6 years, 10 months ago

Did the same

Aditya Kumar - 5 years ago

Masba Islam
May 4, 2014

Python code:

s=0

a=1

for i in range(7917):

a=a%7919

if i==7916:

    a=a

else:

    a=a*(i+2)

print a

Kartik Sharma
Jul 7, 2014

did the same way as Sanchyapol but yes man this is the 1000th PRIME!!!!!

How did you manage to find that out?

Krishna Ar - 6 years, 11 months ago

I just asked "Is 7919 prime" on Google. The problem should be updated to give that information.

Michael Tang - 6 years, 10 months ago