If today is Sunday, then which day is it exactly on 10¹⁰ th day?

To determine the day of the week exactly on the 10^10th day, we need to find the remainder when 10^10 is divided by 7, as days of the week repeat every 7 days.

1. Calculate 10 modulo 7: 10 mod 7 = 3.

2. Therefore, 10^10 mod 7 is equivalent to 3^10 mod 7.

3. Calculate the powers of 3 modulo 7 to find the cycle: 3^1 mod 7 = 3 3^2 mod 7 = 9 mod 7 = 2 3^3 mod 7 = (3^2 * 3) mod 7 = (2 * 3) mod 7 = 6 3^4 mod 7 = (3^3 * 3) mod 7 = (6 * 3) mod 7 = 18 mod 7 = 4 3^5 mod 7 = (3^4 * 3) mod 7 = (4 * 3) mod 7 = 12 mod 7 = 5 3^6 mod 7 = (3^5 * 3) mod 7 = (5 * 3) mod 7 = 15 mod 7 = 1 The cycle length is 6 (since 3^6 mod 7 = 1).

4. Now, simplify 3^10 mod 7 using the cycle: Divide the exponent (10) by the cycle length (6): 10 = 1 * 6 + 4. So, 3^10 mod 7 = (3^6)^1 * 3^4 mod 7 = 1^1 * 3^4 mod 7 = 3^4 mod 7. From step 3, 3^4 mod 7 = 4. Thus, 10^10 mod 7 = 4.

5. Map the remainder to the day of the week. If "today is Sunday", we can assign numerical values to the days starting from Sunday as 1: 1 = Sunday 2 = Monday 3 = Tuesday 4 = Wednesday 5 = Thursday 6 = Friday 0 (or 7) = Saturday

   Since the remainder is 4, the day corresponds to Wednesday.

Analyzing the options: A) Wednesday: Corresponds to a remainder of 4 in our mapping. This matches our calculation. B) Thursday: Corresponds to a remainder of 5. C) Friday: Corresponds to a remainder of 6. D) Saturday: Corresponds to a remainder of 0 (or 7).

The final answer is A) Wednesday.

The final answer is A) Wednesday