The remainder when $10^{10} \cdot (10^{10} + 1) \cdot (10^{10} + 2)$ is divided by $6$ is

The $50^{\text{th}}$ root of a number $x$ is $12$ and the $50^{\text{th}}$ root of another number $y$ is $18$. Then the remainder obtained on dividing $(x + y)$ by $25$ is $........$.

The fractional part of the number $\frac{4^{2022}}{15}$ is equal to

The remainder when $(2023)^{2023}$ is divided by $35$ is $..........$.

The remainder on dividing $5^{99}$ by $11$ is

If $a, b$ and $n$ are natural numbers,then $a^{2n-1} + b^{2n-1}$ is divisible by