Let $N$ be a square number. If its tens digit is odd, then its units digit must be $6$.

Let $N$ be a square number. If its units digit is neither $4$ nor $6$, then its tens digit must be even.

A person eats $X ( > 1)$ cookies in $N$ days in the following way:

• He eats $1$ plus $1/7$ of the remaining cookies on the $1^{st}$ day 
• He eats $2$ plus $1/7$ of the remaining cookies on the $2^{nd}$ day
• $\cdots$
• Finally, he eats the last $N$ cookies on the $N^{th}$ day

What is the smallest possible value of $X$?

Let $n^2$ be a square number. Show that $n^2\equiv 0, \pm 1\pmod{5}$.

Let $n$ be any positive integer, show that $$(5n+1)(5n+2)(5n+3)(5n+4)\equiv -1 \pmod{25}$$

Compute $3^{2018} \mod{17}$.

Compute $\underbrace{3^{3^{3^{\cdots^{3}}}}}_{2012\ times}\pmod{100}$.

Solve $15x\equiv 7\pmod{32}$.

Solve $x^{12}\equiv 3\pmod{11}$.

Show that $x^5\equiv 3\pmod{11}$ is not solvable.

Find one solution to $x^7\equiv 3\pmod{11}$.

Show that $(2^{1194} + 1)$ is a multiple of $65$.

Solve $x^{22} + x^{11}\equiv 2\pmod{11}$.

Compute $20!\pmod{23}$.

Let $p$ be a prime and integer $a$ is co-prime to $p$, show that $$a^{p(p-1)}\equiv 1\pmod{p^2}$$

Let $p$ and $q$ be two distinct primes, and integer $a$ is co-prime to both $p$ and $q$, show $$a^{(p-1)(q-1)}\equiv 1\pmod{pq}$$

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

Given $30!$ ends with some zeros, what is the digit that immediately precedes these zeros?

The two-digit integers from $19$ to $92$ are written consecutively to form the large integer $$N=192021\cdots 909192$$

Suppose that the $3^k$ is the highest power of $3$ that is a factor of $N$. What is $k$.

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.

Let $\mathbb{S}$ be the set of integers between $1$ and $2^{40}$ that contain two $1$s when written in base $2$. What is the probability that a random integer from $\mathbb{S}$ is divisible by $9$?