Number systems

Let the smallest value of $k\in\mathbb N$, for which the coefficient of $x^3$ in $(1+x)^3+(1+x)^4+(1+x)^5+\dots+(1+x)^{99}+(1+kx)^{100}$, $x\ne0$, is $\left(43n+\dfrac{101}4\right)\left({}^{100}C_3\right)$ for some $n\in\mathbb N$, be $p$. Then the value of $p+n$ is:

Let $S = \left\{a + b\sqrt{2} : a, b \in \mathbb{Z}\right\}$, $T_1 = \left\{\left(-1+\sqrt{2}\right)^n : n \in \mathbb{N}\right\}$, and $T_2 = \left\{\left(1+\sqrt{2}\right)^n : n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

The remainder when $3^{2022}$ is divided by 5 is :

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ______.

If $\dfrac1{2\cdot3^{10}}+\dfrac1{2^2\cdot3^9}+\dots+\dfrac1{2^{10}\cdot3}=\dfrac K{2^{10}\cdot3^{10}}$, then the remainder when K is divided by 6 is :

The remainder when $(2021)^{2023}$ is divided by 7 is :

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______.

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is $p$, then $96p$ is equal to ______.

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\dfrac{\sqrt x}{5^{1/4}}+\dfrac{\sqrt5}{x^{1/3}}\right)^{60}$ is $5^k\cdot l$, where $l,k\in\mathbb N$ and $l$ is co-prime to 5, then $k$ is equal to ______.

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^3+\dfrac3{x^k}\right)^{12}$, $x\ne0$ is $2^8\cdot l$, where $l$ is an odd integer, is ______.

The number of values of $a\in\mathbb N$ such that the variance of $3,7,12,a,43-a$ is a natural number is:

The remainder when $(11)^{1011}+(1011)^{11}$ is divided by 9 is

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ______.

Let $f,g:\mathbf N-\{1\}\to\mathbf N$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^\alpha$ divides $a$, and $g(a)=a+1$, for all $a\in\mathbf N-\{1\}$. Then, the function $f+g$ is

The remainder when $(2021)^{2022}+(2022)^{2021}$ is divided by 7 is

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :

Let S = {1, 2, 3, ..., 2022}. Then the probability, that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is :

Let p and p + 2 be prime numbers and let

$$\Delta=\begin{vmatrix}p! & (p+1)! & (p+2)!\\ (p+1)! & (p+2)! & (p+3)!\\ (p+2)! & (p+3)! & (p+4)!\end{vmatrix}$$

Then the sum of the maximum values of α and β, such that $p^\alpha$ and $(p+2)^\beta$ divide Δ, is _______.

Let S = {4, 6, 9} and T = {9, 10, 11, ..., 1000}. If A = {$a_1+a_2+\dots+a_k$ : k ∈ N, $a_1,a_2,a_3,\dots,a_k$ ∈ S}, then the sum of all the elements in the set T − A is equal to _______.

If $p_1$ and $p_2$, respectively, are the second largest and the fourth largest prime numbers in the factorization of the coefficient of $x^{30}$ in the expression of $\left(x^2+2+\dfrac1{x^2}\right)^{-5}(1+x^2)^{40}$, then $p_1+p_2$ is equal to _______