ABC of GCD

Objective

Test your mathematical prowess by solving GCD, LCM, and simplification problems — over the integers, polynomials or rational functions — while five charismatic bunnies try to convince you to pick their answers!

How to Play

1. Choose Your Field: Integers asks about whole numbers like GCD(12, 18). Polynomials asks about polynomials in x, like GCD((x − 1)(x + 2), (x − 1)(x + 3)) = (x − 1). Rational Functions asks about fractions of polynomials R(x) = N(x)/D(x).
2. Factored or Expanded: Choose whether polynomials are shown already factored, or expanded (x² + x − 2) so you have to factor them yourself. Answers are always in factored form.
3. Choose Your Level (1-5): Higher levels have more challenging problems with larger numbers and more prime factors (or more polynomial factors, bigger roots, higher powers and irreducible quadratics like x² + 1).
4. 10 Questions Total:
   • Questions 1-3: Greatest Common Divisor (GCD)
   • Questions 4-6: Least Common Multiple (LCM)
   • Questions 7-10: Simplify Fractions (P(x) / Q(x) in polynomial mode, R₁(x) / R₂(x) in rational-function mode)
5. Meet the Bunnies: Five bunnies will each provide a different answer. Only one is correct!
6. Bunny Drama: Watch as bunnies plead, argue, and sometimes make peace with each other while trying to win your vote.
7. Pick Your Answer: You have 10 to 60 seconds (depending on level) to select the bunny with the correct answer.
8. Learn & Improve: At the end, review your answers with detailed explanations using prime factorization or Euclidean division.

Mathematical Concepts

GCD (Greatest Common Divisor): The largest positive integer that divides both numbers without remainder.

LCM (Least Common Multiple): The smallest positive integer that is divisible by both numbers.

Simplify Fractions: Divide numerator and denominator by their GCD to get the simplest form.

Polynomial GCD / LCM: Works just like prime factorization. Write each polynomial as a product of irreducible factors — (x − r) and (x² + c) — then the GCD takes the common factors with the lowest exponent and the LCM takes every factor with the highest exponent.

Rational Function GCD / LCM: For fractions in lowest terms, GCD(N₁/D₁, N₂/D₂) = GCD(N₁, N₂) / LCM(D₁, D₂) and LCM(N₁/D₁, N₂/D₂) = LCM(N₁, N₂) / GCD(D₁, D₂). To simplify R₁ / R₂, multiply by the reciprocal and cancel common factors.

Tips

• Don't let the bunnies distract you with their antics!
• Use prime factorization to solve complex problems
• In polynomial mode, compare the exponent of each factor — remember (x − 2) and (x + 2) are different factors!
• In expanded mode, find a root r first (try small integers), then divide out (x − r) and repeat.
• Pay attention to the insults—sometimes they reveal clues!
• The timer shows how much time you have left