Practice

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Let $m$ be the least positive integer divisible by $17$ whose digits sum is $17$. Find $m$.

The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.

122

Let $a > b > c$ be three positive integers. If their remainders are $2$, $7$, and $9$ respectively when being divided by $11$. Find the remainder when $(a+b+c)(a-b)(b-c)$ is divided by $11$.

123

Find all positive integer $n$ such that $2^n+1$ is divisible by 3.

167

What is the remainder when $\left(8888^{2222} + 7777^{3333}\right)$ is divided by $37$?

174

Find all prime number $p$ such that both $(4p^2+1)$ and $(6p^2+1)$ are prime numbers.

190

Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:

If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.

208

For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) =\displaystyle\max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$.

219

Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$.

236

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.

250

Let $R$ be the set of all possible remainders when a number of the form $2^n$, where $n$ is a non-negative integer, is divided by $1000$. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by $1000$.

268

There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$, $m$ divides $(a_{n+m} - a_{n})$ for all integers $n$ with $1 \leq n < n+m \leq 30$. Find $N$.

272

What is the last digit of $9^{2019}$?

273

What are the last two digits of $8^{88}$?

274

Find the remainder when $3^{2019} + 4^{2019}$ is divided by 5?

280

Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999}}$ is divided by $1000$.

310

Let integer $a$, $b$, and $c$ satisfy $a+b+c=0$, prove $|a^{1999}+b^{1999}+c^{1999}|$ is a composite number.

311

The number $2^{29}$ is a nine-digit number whose digits are all distinct. Which digit of $0$ to $9$ does not appear?

411

What is the units digit of the sum of the squares of the integers from $1$ to $2015$, inclusive?

474

The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?

991

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?

1117

What are the last two digits in the sum of the factorials of the first $100$ positive integers?

1120

Let four positive integers $a$, $b$, $c$, and $d$ satisfy $a+b+c+d=2019$. Prove $\left(a^3+b^3+c^3+d^3\right)$ cannot be an even number.

1171

What is the tens digit of $7^{2019}$?

1186

What is the units digit of $13^{2019}$?