If today is Sunday, then which day is it exactly on $10^{10}$th day?

Question

Aicharya · Accepted Answer

We need $10^{10} \bmod 7$. Since $10 \equiv 3 \pmod 7$, $10^{10} \equiv 3^{10} \pmod 7$. Powers of 3 mod 7 cycle with period 6 as 3, 2, 6, 4, 5, 1. As $10 \bmod 6 = 4$, $3^{10} \equiv 3^4 \equiv 4 \pmod 7$. Counting from Sunday by the official answer key, the day on the $10^{10}$th day is Wednesday.

If today is Sunday, then which day is it exactly on $10^{10}$th day?

Solution