Reminder when 43^101 + 23^101 is divided by 66
To find the reminder when \(43^{101} + 23^{101}\) is divided by 66, we first need to calculate the individual remainders of each term when divided by 66.
- **Calculating the remainder of 43^101 when divided by 66:**
43 divided by 66 leaves a remainder of 43. Now, let's consider the powers of 43 when divided by 66.
When we divide \(43^1\) by 66, the remainder is 43.
When we divide \(43^2\) by 66, the remainder is 19.
When we divide \(43^3\) by 66, the remainder is 1.
This pattern repeats every 3 powers. Since 101 is not a multiple of 3, the remainder of \(43^{101}\) when divided by 66 will be the same as the remainder of \(43^2\) when divided by 66, which is 19.
- **Calculating the remainder of 23^101 when divided by 66:**
23 divided by 66 leaves a remainder of 23. Now, let's consider the powers of 23 when divided by 66.
When we divide \(23^1\) by 66, the remainder is 23.
When we divide \(23^2\) by 66, the remainder is 37.
When we divide \(23^3\) by 66, the remainder is 1.
This pattern also repeats every 3 powers. Since 101 is not a multiple of 3, the remainder of \(23^{101}\) when divided by 66 will be the same as the remainder of \(23^2\) when divided by 66, which is 37.
Now, adding the remainders of \(43^{101}\) and \(23^{101}\) gives us 19 + 37 = 56.
Therefore, when \(43^{101} + 23^{101}\) is divided by 66, the remainder is 56.