A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes

Let p1, p2,..., pn be distinct prime numbers, and let Nn be their product. We prove that, for any positive integer L that is divisible by the least common multiple of p1 minus one, p2 minus one, and so on, and for integers a1, a2,..., an satisfying that each ai is relatively prime to Nn and shares the same prime factor pi, a certain congruence relation holds among their Lth powers.