Find the remainder when 7¹¹ + 7¹¹¹ + 7¹¹¹¹ is divided by 8.

Question: Find the remainder when 7¹¹ + 7¹¹¹ + 7¹¹¹¹ is divided by 8.

Solution:
When, 7 is divided by 8 then remainder = 7
When, 7² is divided by 8 then remainder = 1
When, 7³ is divided by 8 then remainder = 7
When, 7⁴ is divided by 8 then remainder = 1
Odd exponents give remainder = 7
Even exponents give remainder = 1

In 1st term, exponent is 11, which is odd so remainder = 7
In 2nd term, exponent is 111, which is also odd so remainder = 7
In 3rd term, exponent is 1111, which is odd so remainder = 7
So, (7¹¹ + 7¹¹¹ + 7¹¹¹¹) mod 8 = (7 + 7 + 7) mod 8 = 21 mod 8 = 5