# Questions based on Wilson’s theorem for training for RMO

1(a) Find the remainder when $15!$ is divided by 17.
1(b) Find the remainder when $2(26!)$ is divided by 29.

2: Determine whether 17 is a prime by deciding if $16! \equiv -1 {\pmod 17}$

3: Arrange the integers 2,3,4, …, 21 in pairs a and b that satisfy $ab \equiv 1 {\pmod 23}$.

4: Show that $18! \equiv -1 {\pmod 437}$.

5a: Prove that an integer $n>1$ is prime if and only if $(n-2)! \equiv 1 {\pmod n}$.
5b: If n is a composite integer, show that $(n-1)! \equiv 0 {\pmod n}$, except when $n=4$.

6: Given a prime number p, establish the congruence $(p-1)! \equiv {p-1} {\pmod {1+2+3+\ldots + (p-1)}}$

7: If p is prime, prove that for any integer a, $p|a^{p}+(p-1)|a$ and $p|(p-1)!a^{p}+a$

8: Find two odd primes $p \leq 13$ for which the congruence $(p-1)! \equiv -1 {\pmod p^{2}}$ holds.

9: Using Wilson’s theorem, prove that for any odd prime p:
$1^{2}.3^{2}.5^{2}.\ldots (p-2)^{2} \equiv (-1)^{(p+1)/2} {\pmod p}$

10a: For a prime p of the form $4k+3$, prove that either

$(\frac{p-1}{2})! \equiv 1 {\pmod p}$ or $(\frac{p-1}{2})! \equiv -1 {\pmod p}$

10b: Use the part (a) to show that if $4k+3$ is prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1.

More later,
Nalin Pithwa.