• Stern-Brocot Tree (Visualization : https://sigh.github.io/Stern-Brocot-Tree/)
  • Why each mediant fraction turn out be in lowest terms when it appears in this tree?
  • Why do all possible fractions occur exactly once?
  • Can any fraction be missing from this Tree?
• Introduction to Modular Arithmetic
  • Addition, Subtraction and Multiplication over modulo
  • Why division does not work ?
  • Special Cases of division
• Exercises:
  • Prove that, a number in decimal notation is divisible by 3, if and only if the sum of its digits is divisible by 3.
  • Show that, if $p \text{ mod }4=3$, there is no integer $n$ such that $p$ divides $(n^2+1)$.
  • Prove that, if $n^j\equiv1 \text{ and } n^i \equiv1 (\text{mod } m)$ , then $n^{\text{gcd(i,j)}}\equiv1(\text{mod }m)$.
  • Prove that, $n(n+1)(2n+1)$ is always divisible by 6 for any integer n.