Recreational math word problem with large exponents

Question

Today is Wednesday.  Counting today, what day of the week will be 10²⁰⁰⁹ days from today?

Answer 1

There are 7 days in a week, so the problem can be solved using modulo n arithmetic.

10²⁰⁰⁹ ≡ 3²⁰⁰⁹ (mod 7)

Since 7 is a prime number, 3⁶ ≡ 1 (mod 7) {Fermat's Little Theorem}

3²⁰⁰⁹ = 3²⁰⁰⁴⁺⁵ = (3⁶)³³⁴ 3⁵≡ 3⁵ (mod 7)≡ 5 (mod 7)

The answer is Monday, 5 days from today (Wednesday).

Answer 2

That day would be a Sunday.