Question

Find the remainder when the number represented by 22334 raised to the power (12+22+...+662) is divided by 5?

Answer 1

Question is $\bold{22334^{(1^2+2^2+3^2+4^2+5^2+......66^2)}}$ is divided by 5 , we have to find out remainder .

First of all we have to find out remainder in case of (1² + 2² + 3² + 4² +.... + 66²) are divided by 4 .
we know, 1² + 2² + 3² +... n² = n(n+1)(2n+1)/6
∴ 1² + 2² + 3² + 4² + ...... 66² = 66(66+1)(2×66+1)/6
= 11 × 67 × 133
= 98021
Now, 98021 is divided by 4 leaves 1 as remainder.
∴ you can use $\bold{22334^{(1^2+2^2+3^2+4^2+5^2+......66^2)}}$ is divided by 5 is equivalent 22334 is divided by 5 .
22334 is divided by 5 leaves 4 as remainder.

Hence, answer is 4

Answer 2

Answer:

The given expression can be seen as (22334^ODD POWER)/5, since the sum of 1 ^2 + 2 ^2 + 3 ^2 + 4 ^2 + … + 66 ^2 can be seen to be an odd number.

The remainder would always be 4 in such a case.