Numbers $1,2,\cdots, 1974$ are written on a board. You are allowed to replace any two of these numbers by one number which is either the sum or the difference of these numbers. Show that after $1973$ times performing this operation, the only number left on the board cannot be $0$.

On an $8\times 8$ chess board, there are $32$ white pieces and $32$ black pieces, one piece in each square. If a player can change all the white pieces to black and all the black pieces to white in any row or column in a single move, then is it possible that after finitely many movies, there will be exactly one black piece left on the board?

Find the number of integer pairs $(x, y)$ such that $x^2 + y^2 = 2019$.

If a square number's tens digit is $7$, what is its units digit?

There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.

Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?

Let $n^2$ be a square number, show that $n^2\equiv 0, 1\pmod{4}$.

Show that if $n^2$ is a square number, then $n^2\equiv 0, 1, 4, 9\pmod{16}$.

In plain English, this means that the remainder can only be $0$, $1$, $4$ or $9$ when a square number is divided by $16$.

How many terms in this sequence are squares? $$1, 11, 111, 1111, \cdots $$

How many terms in this sequence are squares? $$4, 44, 444, 4444, \cdots$$

Let $N$ be an odd square number. Show that $N$'s tens digit must be even.

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

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.

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

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 $9$?

Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.

Let $a$ and $b$ be two positive integers such that both of them can be written as a sum of two squares. Show that their product can be written as a sum of two squares in two ways.

(Two Squares Theorem) Show that a positive integer $n$ is a sum of two squares if and only if each prime factor $p$ of $n$ such that $p\equiv 3\pmod{4}$ occurs to an even power in the prime factorization of $n$.

Let $S$ be the sum of squares of $10$ consecutive positive integers. Show $S$ cannot be a square.