Q79. If today is Sunday, then which day is it exactly on 10^10 th day?
Detailed Solution
To find the day on the 10^10th day from Sunday, we need to determine the remainder when 10^10 is divided by 7 (the number of days in a week). This remainder will tell us how many days after Sunday the target day falls.
First, find the remainder of 10 when divided by 7: 10 mod 7 = 3. So, we need to calculate 3^10 mod 7. Let's find the pattern of powers of 3 modulo 7: 3^1 ≡ 3 (mod 7) 3^2 ≡ 9 ≡ 2 (mod 7) 3^3 ≡ 3 × 2 ≡ 6 (mod 7) 3^4 ≡ 3 × 6 ≡ 18 ≡ 4 (mod 7) 3^5 ≡ 3 × 4 ≡ 12 ≡ 5 (mod 7) 3^6 ≡ 3 × 5 ≡ 15 ≡ 1 (mod 7) The cycle length is 6.
Now, divide the exponent 10 by 6: 10 = 1 × 6 + 4. So, 3^10 ≡ (3^6)^1 × 3^4 ≡ 1^1 × 3^4 ≡ 3^4 ≡ 4 (mod 7). The remainder is 4. Starting from Sunday, the 4th day after Sunday is Thursday (Sunday + 1 = Monday, +2 = Tuesday, +3 = Wednesday, +4 = Thursday).
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