[b]Q. [/b] 'm' is the remainder when \(3^{2020}\) is divided by 13 and 'n' is the remainder when \(3^{2021}\) is ...

[Max@Math Revolution]

Q. (Number) 'm' is the remainder when \(3^{2020}\) is divided by 13 and 'n' is the remainder when \(3^{2021}\) is divided by 13. What is the value of m + n?

A. 6
B. 8
C. 10
D. 12
E. 14

[Wendy950612]

3^2019=27^673=(26+1)^673=26A+1^673
3^2019/3=(26A+1^673)/13 thus the remainder is 1
3^2020=3*3^2019 thus the remainder is 3
3^2021=9*3^2019 thus the remainder is 9

so m+n=12

[Max@Math Revolution]

Solution:

Let’s see the remainder pattern when 3 and its higher powers are divided by 13.

=> Since 3 = 13*0 + 3, when 3 is divided by 13, we get a remainder of 3.
=> Since 9 = 13*0 + 9, when 9 is divided by 13, we get a remainder of 9.
=> Since 27 = 13*2 + 1, when 27 is divided by 13, we get a remainder of 1.
=> Since 81 = 13*6 + 3, when 81 is divided by 13, we get a remainder of 3.

So, the pattern of the remainders is 3, 9, 1, 3, 9, 1…

'm' is the remainder when \(3^{2020}\) is divided by 13. Since 2020 = 673*3 + 1, we get that the remainder when \(3^{2020}\) is divided by 13 is equal to the remainder when 31 is divided by 13. Therefore, we get a remainder of 3 when \(3^{2020}\) is divided by 13 and m = 3.

'n' is the remainder when \(3^{2021}\) is divided by 13. Since 2021 = 673*3 + 2, we get that the remainder when \(3^{2021}\) is divided by 13 is equal to the remainder when 32 is divided by 13. Therefore, we get a remainder of 9 when \(3^{2021}\) is divided by 13 and n = 9.

Therefore, m + n = 3 + 9 = 12 and D is the correct answer.

Answer D