Wilson's theorem
- Olympiad · IOQM
Wilson's theorem says that a whole number p greater than 1 is prime exactly when p divides (p - 1) factorial plus 1.
Here n! (read n factorial) means 1 × 2 × 3 × ... × n. So 4! = 24 and 6! = 720. In the language of congruence, the theorem says (p - 1)! ≡ -1 (mod p) when p is prime.
The rule works both ways. If p is prime, the remainder is always p - 1. If p is not prime, it never is. So in theory it is a full test for primes. In practice factorials grow far too fast for large numbers. Its real use is in proofs and remainder questions.
A small example
- p = 5: 4! + 1 = 24 + 1 = 25, and 25 = 5 × 5. It works, and 5 is prime.
- p = 7: 6! + 1 = 720 + 1 = 721, and 721 = 7 × 103. It works, and 7 is prime.
- p = 6: 5! + 1 = 120 + 1 = 121. Since 121 = 6 × 20 + 1, 6 does not divide it. And 6 is not prime.
A question may ask for the remainder of 10! on division by 11. Since 11 is prime, 10! ≡ -1 (mod 11). So the remainder is 11 - 1 = 10.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under number theory. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Number Theory) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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