I cannot prove that: If P is a prime number, and Q < P (or Q is not a multiple of P), then Q^(P-1) -1 must be a multiple of P.

I don't know. About two years ago, I unintentionally found that for 3, 5 and 7 the statement is true. At first I guess it is true for all odd numbers. Then I tried for 9, 11,13, 15, and found that it is not true for 9 and 15. Then in subsequent two years I tried two more numbers 17 and 19, they are both true. Hence I guess that it is true for all prime numbers. However, I just cannot prove it.