The remainder when $3 ^ { 2022 }$ is divided by 5 is
(1) 1
(2) 2
(3) 3
(4) 4

The remainder when $(11)^{1011} + (1011)^{11}$ is divided by 9 is $\_\_\_\_$.
(1) 1
(2) 8
(3) 6
(4) 4

Let the number $( 22 ) ^ { 2022 } + ( 2022 ) ^ { 22 }$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7 . Then $\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$ is equal to
(1) 20
(2) 13
(3) 5
(4) 10

The remainder on dividing $5^{99}$ by 11 is $\underline{\hspace{1cm}}$.

The remainder, when $7 ^ { 103 }$ is divided by 23 , is equal to :
(1) 6
(2) 17
(3) 9
(4) 14

$$\begin{aligned} & 2 ^ { x } \equiv 1 ( \bmod 7 ) \\ & 3 ^ { y } \equiv 4 ( \bmod 7 ) \end{aligned}$$
For the smallest positive integers x and y satisfying these congruences, what is the difference $y - x$?
A) 5
B) 4
C) 3
D) 2
E) 1

$$\lim _ { x \rightarrow \infty } \frac { e ^ { - 3 x } + e ^ { 2 x } } { \ln x + 3 e ^ { 2 x } }$$
What is the value of this limit?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 1 } { 3 }$
D) 0
E) 1

$$1 ^ { 5 } + 2 ^ { 5 } + 3 ^ { 5 } + 4 ^ { 5 } + 5 ^ { 5 }$$
What is the remainder when this expression is divided by 7?
A) 4
B) 3
C) 2
D) 1
E) 0

Let $a$ and $b$ be natural numbers such that $$\begin{aligned}& 4 \cdot a \equiv 2 ( \bmod 11 ) \\& 4 \cdot b \equiv 5 ( \bmod 7 )\end{aligned}$$ the following congruences are given.\ Accordingly, what is the smallest value that the sum $\mathbf{a+b}$ can take?\ A) 7\ B) 9\ C) 11\ D) 13\ E) 15