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.