The number of positive integers $n$ less than or equal to 22 such that 7 divides $n ^ { 5 } + 4 n ^ { 4 } + 3 n ^ { 3 } + 2022$ is
(A) 7
(B) 8
(C) 9
(D) 10

Suppose $x$ and $y$ are positive integers. If $4 x + 3 y$ and $2 x + 4 y$ are divided by 7, then the respective remainders are 2 and 5. If $11 x + 5 y$ is divided by 7, then the remainder equals
(A) 0.
(B) 1.
(C) 2.
(D) 3.

Let $N$ be a 50 digit number. All the digits except the 26th one from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(a) 1 .
(B) 3 .
(C) 7 .
(D) 9 .

If $(27)^{999}$ is divided by 7, then the remainder is
(1) 3
(2) 1
(3) 6
(4) 2

Let $A = \{ n \in N : n$ is a 3-digit number $\}$, $B = \{ 9 k + 2 : k \in N \}$ and $C = \{ 9 k + l : k \in N \}$ for some $l ( 0 < l < 9 )$. If the sum of all the elements of the set $A \cap ( B \cup C )$ is $274 \times 400$, then $l$ is equal to

The total number of two digit numbers $n$, such that $3 ^ { n } + 7 ^ { n }$ is a multiple of 10, is $\underline{\hspace{1cm}}$.

If the remainder when $x$ is divided by 4 is 3, then the remainder when $( 2020 + x ) ^ { 2022 }$ is divided by 8 is $\underline{\hspace{1cm}}$.

$3 \times 7 ^ { 22 } + 2 \times 10 ^ { 22 } - 44$ when divided by 18 leaves the remainder

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

The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is
(1) 0
(2) 2
(3) 3
(4) 4

The remainder when $( 2021 ) ^ { 2023 }$ is divided by 7 is
(1) 2
(2) 3
(3) 4
(4) 5

The remainder when $( 2021 ) ^ { 2022 } + ( 2022 ) ^ { 2021 }$ is divided by 7 is
(1) 0
(2) 1
(3) 2
(4) 6

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 when $19^{200} + 23^{200}$ is divided by 49, is $\_\_\_\_$.

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

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

Q82. The remainder when $428 ^ { 2024 }$ is divided by 21 is

$$\begin{aligned} & 2 ^ { x } \equiv 1 ( \bmod 7 ) \\ & 3 ^ { y } \equiv 4 ( \bmod 7 ) \end{aligned}$$
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