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Factor Fortress: GCF & LCM Siege

Understand factors and multiples through mechanical gear synchronization, then defend the castle against siege waves!

Step 1: The Briefing

GCF vs. LCM: The Division & Synchronization Twins

Many adventurers mix up GCF and LCM. Here is the gamer-proof way to master them forever:

GCF (Greatest Common Factor): The "Even Loot Splitter"

What is the largest number that divides into both values evenly with zero remainders? Use GCF when cutting materials or distributing inventory.

LCM (Least Common Multiple): The "Cooldown Synchronizer"

What is the first future number that appears on both multiplication tables? Use LCM to find when two cycles or timers hit at the exact same moment.

Step 2: The Lab Bench

Dual-Gear Mechanical Synchronizer

Adjust the teeth on Gear A and Gear B. Watch how GCF and LCM change live as the gears spin!

6 Teeth
8 Teeth
Least Common Multiple (LCM)
24

Both white alignment dots meet after exactly this many teeth pass the threshold!

Greatest Common Factor (GCF)
2

Largest integer that divides both gear teeth counts evenly.

Step 3: Quick Warm-Up

Fortress Cadet Drill

Archer A fires every 3 seconds. Archer B fires every 5 seconds. When will they fire together for the first time? (LCM)

DIFFICULTY:
Step 4: The Arcade Arena Wave 1: Dual Catapult Sync
Siege Barrier Timer
Siege Catapult A reloads every 4 seconds. Catapult B reloads every 6 seconds. When will they BOTH fire at the exact same moment? (LCM)
Calculate GCF or LCM before the enemy reaches the gate!
Step-by-step lesson

Factors and multiples: sharing equally and matching cycles

Before you start: Recall multiplication tables and divide whole numbers exactly.

Distinguish a factor from a multiple

A factor divides a whole number without a remainder: 3 is a factor of 18 because 18 = 3 × 6. A multiple is obtained by multiplying: 36 is a multiple of 18 because 36 = 18 × 2. The GCF is the greatest common factor; GCD is another name for the same value. The LCM is the least positive common multiple.

Worked example: split two supplies into identical packs

You have 18 blue tokens and 24 gold tokens. To make the largest possible number of identical packs using every token, find GCF(18, 24). Their prime factorizations are 18 = 2 × 3² and 24 = 2³ × 3. A common divisor can use only the primes present in both, and only as many copies as both supply. Choose the smaller powers: 2 × 3 = 6 packs. Each contains 3 blue and 4 gold tokens.

Worked example: match two repeating timers

If timers restart every 18 and 24 seconds and start together, when do they next restart together? A shared time must contain enough prime factors to be a multiple of both intervals. Choose the larger powers: LCM = 2³ × 3² = 72 seconds. Check: 72 ÷ 18 = 4 and 72 ÷ 24 = 3.

Why the smallest and largest powers differ

A common factor must fit inside each original number, so it cannot take more copies of a prime than either has. A common multiple must contain both factorizations, so it needs enough copies for the larger requirement. Multiplying 18 × 24 gives a common multiple, 432, but counts shared factors twice. For positive integers, GCF × LCM = 18 × 24: here 6 × 72 = 432.

If the GCF is 1, the numbers are coprime. They need not both be prime: 8 and 9 are coprime, and their LCM is 72.

Try it yourself

Find the GCF and LCM of 12 and 20. Check using their product.

Show answer and explanation

12 = 2² × 3 and 20 = 2² × 5. The GCF is 4; the LCM is 60. The check is 4 × 60 = 240 = 12 × 20.

Use the activity to check your understanding

Decide whether a challenge asks for an equal partition or a shared cycle before choosing GCF or LCM.

Try the next activity: Adding Fractions →

Sources and further reading