Category Archives: Number Theory

How many pairs of positive integers $(x,y)$ are there such that $x<y$ and $\dfrac{x^2+y^2}{x+y}$ is an integer which is a divisor of $2835$? BIMC 2018 Click for the solution Solution: Based on the result of Number Theory Challenge – 1, … Continue reading →

Let $n=4k+3$ is a prime number and $a^2+b^2\equiv 0\pmod{n}$, prove that $a\equiv b \equiv 0\pmod{n}$. Click for the solution Proof: If $a\equiv 0\pmod{n}$, because $a^2+b^2\equiv 0\pmod{n}$, $b\equiv 0\pmod{n}$. If $a\not\equiv 0\pmod{n}$, because $n$ is a prime number, according to Fermat’s … Continue reading →