Share 12 candies equally with none left over — does any group size work?
A divisor divides a number cleanly; a multiple is the same fact seen the other way — the rules of dividing evenly solve both sharing and meeting problems.
Experiment
Hands-on experiment
🔮 Predict first — how many group sizes can share 12 candies equally with none left over?
🍬 Pick a group size and divide
Split 12 among that many people — does it come out even?
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Division doesn't always work — 12 candies won't split equally among 5. A rule for 'when does it divide cleanly' was needed in advance.
That rule is the divisor. Know 12's divisors (1,2,3,4,6,12) and every possible fair split is visible at once. For two quantities together, common divisors — and the greatest of them, the GCD.
Reversed, 'when do cycles meet again?' is a multiples question. Cycles of 4 and 6 meet at common multiples (12, 24, …), first at the least one — the mathematics of calendars, gears, and bus schedules.
Insight
Insights from the video
“Divisor and multiple are one fact, two directions.”
'3 divides 12' and '12 is a multiple of 3' are the same sentence (12 = 3×4). Looking up from below: divisor. Looking down from above: multiple.
“GCD answers sharing; LCM answers meeting.”
Want the largest fair split? GCD. Want the first moment different cycles align? LCM — two everyday questions, answered exactly.
Misconception
Common misconceptions
Bigger numbers have more divisors.
13 is bigger than 12 but has only 1 and 13 (it's prime). 12 has six: 1,2,3,4,6,12. The divisor count is set not by size but by which primes build the number.
Two cycles re-meet at the product of the two numbers (4 and 6 days → day 24).
Day 24 is a meeting, but the FIRST meeting is day 12. Because 4 and 6 share the factor 2 — lcm = (4×6)÷2 = 12. The product can overshoot.
Formula
Writing it as math
What the candy sharing confirmed, in mathematical language.
Divisors and multiples
3 divides 12 cleanly (a divisor); 12 is a multiple of 3 — one multiplication states both relations.
Greatest common divisor
The largest number dividing both — the maximum group that shares 12 candies and 18 chocolates with none left.
Least common multiple
Where two cycles first re-meet. Divide the product by the GCD — counting the shared factor only once.
In Real Life
Where you meet it in real life
Bus schedules
Buses every 15 and every 9 minutes leaving together will next leave together in lcm(15,9) = 45 minutes.
Gears and bicycles
Mesh gears of 12 and 18 teeth and gcd·lcm set the rotation cycle. The rarer the same teeth meet (coprime counts), the more evenly they wear.
Calendar cycles
Weeks of 7 sliding against months of 30–31; the 12 zodiac animals and 10 stems making the 60-year cycle — lcm(12,10)=60.
Fraction arithmetic
The common denominator is an LCM; simplifying uses the GCD — divisors and multiples are the engine inside fraction arithmetic.
Try Yourself
Test yourself
Q1Find all divisors of 20.Show answer ▾
1, 2, 4, 5, 10, 20 — six of them. The trick is pairing: 1×20, 2×10, 4×5. Divisors always come in multiplication pairs.
Q224 pencils and 36 erasers — the largest group that shares both with none left over?Show answer ▾
gcd(24, 36) = 12 people, each getting 2 pencils and 3 erasers — the GCD gives the maximum for fair splitting.
Q3Swimming every 3 days, hiking every 5 — how often do they land on the same day?Show answer ▾
Every lcm(3, 5) = 15 days. 3 and 5 share no factor but 1 (coprime), so the lcm is simply the product — 'just multiply' is only right when the numbers are coprime.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
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Connection
Concepts connect
Previous concept
Leads to next
Prime Numbers
Instead of hunting divisors one by one, factor the number into primes and every divisor falls out — meet the atoms of multiplication.
Go to the Prime Numbers lab →Related
Labs worth exploring together
Related lab
Prime Numbers
Prime factorization automates divisor-finding — 12=2²×3 computes all six divisors.
Go to the Prime Numbers lab →Related lab
Fractions
Common denominators (LCM) and simplifying (GCD) — the engine of fraction arithmetic lives here.
Go to the Fractions lab →