Elementary Math Word Problems · Number Theory Foundations
GCF Tiling & LCM Bus Dispatch Visualizer
Key pedagogical takeaway: Tiling rooms and cutting boards into maximal pieces uses GCF. Periodic re-alignment of buses and meshing gears uses LCM. GCF * LCM strictly equals a * b.
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3D Simulation Stage● 60 FPS
Parameter Controls
steps
Live Derivations & Metrics
Greatest Common Factor GCF
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Least Common Multiple LCM
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Minimal Square Tiles Count
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GCF x LCM Product Identity
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Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Worked Problem
Problem: Room is 18m by 12m. Covered by largest identical whole square tiles without cutting. Find max tile size and count.
Step-by-Step Derivation:
- Max tile edge = GCF(18, 12) = 6m. Rows = 12/6 = 2, Columns = 18/6 = 3. Total tiles = 2 * 3 = 6 tiles.
Worked Problem
Problem: Bus 1 departs every 10 min; Bus 2 every 15 min. Both depart together at 7:00 AM. When do they next depart together?
Step-by-Step Derivation:
- Interval = LCM(10, 15) = 30 min. Next simultaneous departure = 7:00 AM + 30 min = 7:30 AM.
Student Inquiry Worksheet & Practice Problems
Q1. Variant 1: Two steel pipes (24m and 32m) cut into identical maximum segments without waste. Segment length and total count?
Answer Key: Max segment = GCF(24, 32) = 8m. Segments = 24/8 + 32/8 = 3 + 4 = 7 segments.
Q2. Variant 2: Track runner A laps in 80s, runner B in 100s. When do they next meet at the start line? How many laps each?
Answer Key: Time = LCM(80, 100) = 400s. Runner A completes 400/80 = 5 laps; Runner B completes 400/100 = 4 laps.
Q3. Variant 3: Two numbers have GCF = 6 and LCM = 72 (neither is 6 or 72). Find the two numbers.
Answer Key: A*B = GCF*LCM = 432. A=6x, B=6y with gcd(x,y)=1. x*y = 12. Coprime pair (3,4) yields numbers 18 and 24.
Frequently Asked Questions & Misconceptions
Does GCF(A,B) * LCM(A,B) = A * B hold universally?
Strictly for any TWO positive integers! GCF takes minimum prime powers; LCM takes maximum prime powers. Their product exactly reconstructs A*B. (Fails for 3+ numbers).
If two numbers are coprime (e.g. 8 and 9), what are their GCF and LCM?
GCF = 1. LCM is directly their product A * B (8 * 9 = 72).
How does the Euclidean Algorithm rapidly compute GCF of massive integers?
Repeatedly replaces larger number with remainder: GCF(A, B) = GCF(B, A mod B) until remainder reaches 0. The final non-zero divisor is the GCF.
Gears with 30 teeth and 12 teeth mesh. How many teeth pass before marked teeth re-align?
LCM(30, 12) = 60 teeth! The 30-tooth gear rotates 2 revolutions, and the 12-tooth gear rotates 5 revolutions.