Modular Arithmetic: all problems

Find the remainder when $36^9$ is divided by $5$.

Find the remainder when $34^9$ is divided by $5$.

Find the remainder when $7^{803}$ is divided by $400$.

An integer $N$ is selected at random in the range $1 \le N \le 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?

If $p$, $q$ and $r$ are primes with $pqr = 7(p + q + r)$, find $p + q + r$.

Find the smallest positive number that leaves a remainder of $11$ when divided by $13$ and a remainder of $2$ when divided by $5$.

What is the hundreds digit of $(20! - 15!)$?

What is the tens digit of $2015^{2016} - 2017$?

Which of the following expressions is never a prime number when $p$ is a prime number?

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?

Let $a_1, a_2, \dots, a_{2018}$ be a strictly increasing sequence of positive integers such that

What is the remainder when $a_1^3 + a_2^3 + \cdots + a_{2018}^3$ is divided by $6$?

How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$?

Let $N = 123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?

One of the following numbers is not divisible by any prime number less than $10$. Which is it?

Let $S$ be the set of all integers $z > 1$ such that for all pairs of nonnegative integers $(x, y)$ with $x < y < z$, the remainder when $2025x$ is divided by $z$ is less than the remainder when $2025y$ is divided by $z$. What is the sum of the elements of $S$?

Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^4 + 1$ is divisible by $p^2$. Find the least positive integer $m$ such that $m^4 + 1$ is divisible by $p^2$.

Let $x_0, x_1, x_2, \dots$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define

Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \ge 1$. What is the value of the sum

What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$?

What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?

Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q + r$ divisible by $11$?

How many distinct four-digit numbers are divisible by $3$ and have $23$ as their last two digits?

For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.

Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

It is known that, for all positive integers $k$,

Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \cdots + k^2$ is a multiple of $200$.

Find the sum of all positive integers $n$ such that when $1^3 + 2^3 + 3^3 + \cdots + n^3$ is divided by $n + 5$, the remainder is $17$.

The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?