Final answer:
Using the properties of congruence, we can say that 6 is congruent to -1 modulo 7. This simplifies the expression 6^63 to (-1)^63. Since we only consider positive remainders, the remainder of 6^63 divided by 7 is 6.
Step-by-step explanation:
The question is asking us to find the remainder of 663 divided by 7 using the properties of congruence. We use the property that if a ≡ b (mod m), then an ≡ bn (mod m). So first notice that 6 ≡ -1 (mod 7). Using this property, we can transform 663 into (-1)63 which is congruent to -1 modulo 7. When we divide -1 by 7, the remainder is 6, because we need to consider only the positive remainder. Therefore, the remainder of 663 divided by 7 is 6.
Learn more about Properties of Congruence