Number Theory

Let $x,y,z$ be distinct integers. $x$ and $y$ are odd positive integers and $z$ is an even positive integer. Which of the following cannot be true?

If both $7^2$ and $3^3$ are factors of $a\times11^3\times6^2\times13^{11}$, then the smallest possible value of $a$ is

The number of ordered pairs $(m,n)$, $m,n\in{1,2,\ldots,100}$ such that $7^m+7^n$ is divisible by $5$ is

Find the unit digit of $(13647)^{3265}$.

The sum of the numbers from 1 to 100 which are not divisible by 3 and 5 is:

If $P = {(4n - 3n - 1) : n \in N}$ and $Q = {(9n - 9) : n \in N}$, then $P \cup Q$ equals to:

A treasure chest has less than 100 gold coins. The number of coins is How many coins are there in the chest?

The number if non –negative integers less than 1000 that contain the digit 1 are.

If $x, y, z$ are three consecutive positive integers, then $log (1 + xz)$ is

$a, b, c$ are positive integers such that $a^{2}+2b^{2}-2bc=100$ and $2ab-c^{2}=100$. Then the value of $\frac{a+b}{c}$ is

The remainder when 2³¹ is divided by 5 is

How many 3-digit numbers divisible by 5, can be formed using the digits 2 3 5 6 7 and 9, without repetition of digits?

How many natural numbers smaller than  can be formed using the digits 1 and 2 only?

Two numbers $a$ and $b$ are chosen are random from a set of the first 30 natural numbers, then the probability that $a^2 - b^2$ is divisible by 3 is

If $A = \{4^x- 3x - 1 : x ∈ N\}$ and $B = \{9(x - 1) : x ∈ N\}$, where N is the set of natural numbers, then

Choose the correct option for the remainder when X = 1! + 2! + 3! + ...........+ 100! is divided by 24

If a number x is selected at random from natural numbers 1,2,…,100, then the probability for $x+\frac{100}{x}{\gt}29$ is

If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3 is

Let Z be the set of all integers, and consider the sets $X=\{(x,y)\colon{x}^2+2{y}^2=3,\, x,y\in Z\}$ and $Y=\{(x,y)\colon x{\gt}y,\, x,y\in Z\}$. Then the number of elements in $X\cap Y$ is:

Let $A=\{{5}^n-4n-1\colon n\in N\}$ and $B=\{{}16(n-1)\colon n\in N\}$ be sets. Then

The number of all even integers between 99 and 999 which are not multiple of 3 and 5 is

A group of 630 children is arranged in rows for a group photograph. Each row contains three fewer children than the row in front of it. What number of rows is not possible?