Problem:
Prove that for p > 3 prime, p^2 \equiv 1 \pmod{24} .
Solution:
Consider p^2 - 1 = (p-1) (p+1) .
Since p > 3, we know that either p-1 or p+1 is divisible by 3. Both are divisible by 2. Also, one of them is divisible by 4, since p is either -1 or 1 \pmod 4 . Hence p^2-1 is divisible by 2 \times 3 \times 4 = 24 .
QED
Comments
Post a Comment