Category Archives: Number Theory

Prove that for every positive integer $n$, there is a positive integer $m$ so that $2^n | (19^m-97)$.🔑 Proof: We prove it by induction. Base Case: For $n=1,2,3$, we have $$n=1, m=1 \Longrightarrow 2^1|(19^1-97)$$ $$n=2, m=2 \Longrightarrow 2^2|(19^2-97)$$ $$n=3,m=2 \Longrightarrow … Continue reading →

Prove that for integers $a$,$b$,$c$, if $9|(a^3+b^3+c^3)$, then at least of them is divisible by $3$.🔑 Proof: Let $a=3A+x$, $b=3B+y$, $c=3C+z$, where $A$, $B$, $C$ are integers, $a\pmod 3=x$, $b\pmod 3=y$, $c\pmod 3=z$. Therefore $0\le x,y,z\le 2$. Without loss of … Continue reading →

Let $n=pqrc$, where $p$, $q$, and $r$ are three distinct prime numbers, $p<q<r$, and $c$ is a positive integer. For any two distinct integers $1\le x<y\le n-1$, there exists $s$ which is a proper factor of $n$, $1<s<n$, such that … Continue reading →

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 →