HCF and LCMExtendabout 60 min

Cycles, gears and puzzles: HCF and LCM in the wild

Calendars, cicadas, tabla, bicycles, jugs, screens and olympiad problems

Expeditions beyond the textbook: cycles with head starts, calendars and planetary alignments (and why they are not LCMs), prime-cycle cicadas, gears and bicycle chains, tala rhythms, water jugs, ancient remainder puzzles, screen ratios, fractions, olympiad problems, careers and open questions.

Start at chapter 1

In this part you’ll

Decide whether two cycles with different starting points can ever coincide, using the HCF.
Apply HCF and LCM to gears, rhythms, screens, calendars and jug puzzles, and explain where the models break down.
Solve remainder puzzles, including Sunzi’s, using the LCM and systematic listing.
Find the HCF and LCM of fractions and use them in problems.
Tackle olympiad-style problems that combine HCF, LCM and counting.

HCF and LCM are not just exam topics. They are the mathematics of cycles that repeat and things that must fit exactly, and the world is full of both: calendars and festivals, planets and insects, gears and bicycle chains, tabla rhythms, screen shapes, water jugs, and the codes that protect online payments.

This layer is a set of expeditions. Some are real-world contexts with honest caveats. Some are puzzles, including olympiad-style problems. Some end in questions nobody has answered yet. Pick the ones that interest you; you do not have to go in order.

Words to know

All maths vocabulary →

Words from the expeditions

cycle / period: The length of time (or count) after which something repeats.; Example: A 4-year Olympic cycle
offset (head start): How far apart two cycles start.; Example: FIFA World Cup is offset 2 years from the Olympics.
synodic period: How often two orbiting bodies line up again as seen from the Sun (or Earth).; Example: Jupiter–Saturn: about 19.9 years
Metonic cycle: 19 years, almost exactly 235 lunar months: the basis of many lunisolar calendars.; Example: 19 × 365.24 ≈ 235 × 29.53 days
hunting tooth: An extra gear tooth that makes the two tooth counts co-prime, so every tooth meets every other.; Example: 12 and 17 teeth
tala / sam: A rhythmic cycle in Indian classical music, and its first beat.; Example: Teentaal has 16 beats.
polyrhythm: Two or more different beat cycles played at the same time.; Example: 3 against 2 repeats every 6 pulses.
aspect ratio: The shape of a rectangle as width : height in simplest form.; Example: 1920 × 1080 → 16 : 9
Chinese Remainder Theorem: For pairwise co-prime divisors, any set of remainders has exactly one solution below their LCM.; Example: Sunzi: 23
inclusion–exclusion: Count "a or b" as count(a) + count(b) − count(both).; Example: 250 + 166 − 83 = 333

Chapter 01

Calendars, festivals and cycles with a head start

Suppose a village has a weekly bazaar every 7 days and a travelling cattle fair that returns every 5 days. If both happen today, they coincide again in LCM(7, 5) = 35 days. So far, so Discover.

Real cycles, though, often do not start together. Here is a case that surprises most people. The Summer Olympics are held in years divisible by 4 (2024, 2028, 2032 …). The men's FIFA World Cup is also every 4 years, but in the years in between (2026, 2030, 2034 …). Both have a 4-year cycle, LCM(4, 4) = 4, yet they never fall in the same year. Their head starts differ by 2 years, and no amount of waiting fixes that.

The rule for cycles with a head start: two cycles of lengths a and b can only ever meet if the difference in their starting points is a multiple of HCF(a, b). If they meet once, they then meet every LCM(a, b).

Worked example

0 / 5 steps shown

Two festivals with a head start

A dance festival is held every 4 years starting in 2024. A science fair is held every 6 years starting in 2026. In which year do both first happen? How often after that?

Example

A calendar built on an almost-common multiple

Lunisolar calendars, such as Hindu calendars and the Hebrew calendar, have to keep months (set by the Moon) in step with years (set by the Sun). A month is about 29.53 days and a year about 365.24 days. They have no exact common multiple, but 19 years ≈ 6,939.6 days and 235 months ≈ 6,939.7 days, which agree to within about two hours. This near-match is the Metonic cycle. The Hebrew calendar uses it directly, adding 7 leap months every 19 years. Hindu lunisolar calendars also add an extra month, the adhik maas, but they work it out from the real positions of the Sun and Moon rather than from the 19-year cycle: it comes about once every 32 to 33 months, which works out at roughly seven extra months in 19 years. Real calendars like this are built on "close enough" LCMs.

Chapter 02

Planets and cicadas: nature’s cycles

Here is a curiosity that appears in many puzzle books: "Three planets orbit a star in 3, 4 and 6 years. They are lined up today. When will they line up again?" The textbook answer is LCM(3, 4, 6) = 12 years, because then each planet has made a whole number of orbits (4, 3 and 2) and is back at its starting point.

That is correct for the puzzle, but it answers a slightly different question from real astronomy. In the puzzle, every planet returns to its starting position. Real alignments only need the planets to line up in the same direction from the Sun, anywhere around the orbit, and that happens much more often.

It’s more subtle

Why do some cicadas count in primes?

In eastern North America, periodical cicadas spend 13 or 17 years underground, then emerge together in their millions. Both are prime numbers. One well-known hypothesis says a prime cycle helps them avoid predators that boom on shorter cycles.

The LCM shows the idea. A 12-year cicada would meet a predator with a 2-, 3-, 4- or 6-year cycle every 12 years, at every emergence. A 17-year cicada meets a 2-year predator only every 34 years, a 3-year one every 51 years, and a 6-year one every 102 years.

This is a hypothesis, not a settled result, and biologists still debate it. A second leading idea is that prime cycles stop broods of different cycle lengths from emerging together and interbreeding, which would blur the cycle. It is a great example of mathematics suggesting an explanation that scientists must then test.

Lab

Race cycles of different lengths, from toy planets to tala rhythms, and predict when they first coincide.

Round 1 / 10★ 0 ptsBest: 0

Numbers: 3, 4 and 6

Text version of this activity

This lab races two or three cycles, shown as frogs hopping along a number line. The first shared stone is when the cycles coincide.

Toy planets: cycles 3 and 4 and 6 → first coincide at 12.
12-year cicada, 4-year predator: cycles 12 and 4 → first coincide at 12.
12-year cicada, 6-year predator: cycles 12 and 6 → first coincide at 12.
Teentaal (16 beats) and Jhaptaal (10 beats): cycles 16 and 10 → first coincide at 80.
Ektaal (12 beats) and Rupak (7 beats): cycles 12 and 7 → first coincide at 84.

A cycle with many factors (12) meets short cycles at every step; co-prime cycles such as 12 and 7 take the full product to meet.

Lab

Compare prime and non-prime cicada cycles with predator cycles and see how rarely prime cycles coincide.

Round 1 / 10★ 0 ptsBest: 0

Numbers: 17 and 4

Text version of this activity

This lab lists multiples and shows the prime factors of a cicada cycle and a predator cycle.

17-year cicada, 4-year predator: HCF 1, they coincide every 68 years, which is every 4 cicada emergences.
13-year cicada, 6-year predator: HCF 1, they coincide every 78 years, which is every 6 cicada emergences.
17-year cicada, 6-year predator: HCF 1, they coincide every 102 years, which is every 6 cicada emergences.
12-year cicada, 4-year predator: HCF 4, they coincide every 12 years, which is every 1 cicada emergence.
12-year cicada, 6-year predator: HCF 6, they coincide every 12 years, which is every 1 cicada emergence.

A prime cycle shares no factor with any shorter cycle, so the LCM is the full product and meetings are rare. The 12-year cycle meets 4- and 6-year predators at every single emergence.

Chapter 03

Gears, chains and the hunting tooth

When two gears mesh, their teeth take turns. Paint one tooth on each gear red, and start with the two red teeth touching. When do they touch again?

Every time a tooth passes the meeting point, both gears move on by one tooth. The small gear's red tooth is back after every 12 teeth (for a 12-tooth gear); the big gear's after every 18. They are both back together after LCM(12, 18) = 36 teeth, which is 3 turns of the small gear and 2 turns of the big one.

Engineers care about this. If the same pairs of teeth meet over and over, any tiny flaw on one tooth wears the same partner teeth again and again. Choosing tooth counts that are co-prime, for example 12 and 17, makes every tooth on one gear meet every tooth on the other before the pattern repeats: LCM(12, 17) = 204 teeth, 17 turns of the small gear. The extra tooth that makes the counts co-prime is traditionally called a hunting tooth.

TableTwo meshing gears: when do the same two teeth meet again? (all computed)
Teeth	HCF	LCM (teeth passed)	Turns of small gear	Turns of big gear	Different partner teeth for each tooth
12 and 18	6	36	3	2	3
12 and 17	1	204	17	12	17
20 and 30	10	60	3	2	3
20 and 31	1	620	31	20	31
15 and 40	5	120	8	3	8
16 and 41	1	656	41	16	41

Worked example

0 / 5 steps shown

The bicycle chain

A bicycle chain has 116 links and runs over a front chainring with 48 teeth. A particular chain link sits on a particular tooth. After how many links have passed will that same link sit on that same tooth again? How many pedal turns is that?

Chapter 04

Rhythm: tabla cycles and polyrhythms

Indian classical music is organised in tala cycles. Teentaal has 16 beats, Jhaptaal 10, Ektaal 12 and Rupak 7. The first beat of each cycle is the sam, where musicians often land together with a flourish.

Imagine a duet where one drummer plays in Teentaal (16) and another in Jhaptaal (10), starting together on sam. Their sams coincide again after LCM(16, 10) = 80 beats: 5 cycles of Teentaal and 8 of Jhaptaal. With Teentaal and Rupak, it takes LCM(16, 7) = 112 beats, because 16 and 7 are co-prime.

In a tihai, a phrase is played three times so that it ends exactly on sam; composers use the arithmetic of cycles to make the landing come out right. And in West African, Latin American and Western music, polyrhythms like "3 against 2" repeat every LCM(3, 2) = 6 pulses, and "4 against 3" every 12.

TableWhen do two tala cycles land on sam together? (all computed)
Tala pair	Beats per cycle	HCF	Sams coincide every	Cycles of each
Teentaal + Jhaptaal	16 and 10	2	80 beats	5 and 8
Teentaal + Ektaal	16 and 12	4	48 beats	3 and 4
Teentaal + Rupak	16 and 7	1	112 beats	7 and 16
Jhaptaal + Ektaal	10 and 12	2	60 beats	6 and 5
Ektaal + Rupak	12 and 7	1	84 beats	7 and 12

Chapter 05

Water jugs and the power of the HCF

A classic puzzle: you have an unmarked 3-litre jug and an unmarked 5-litre jug, and a tap. You may fill a jug, empty a jug, or pour from one into the other until one is full or the other empty. Can you measure exactly 4 litres?

Yes, and the Deepen layer explains why: HCF(3, 5) = 1, and any amount that is a multiple of the HCF (up to the bigger jug) can be made. With a 4-litre and a 6-litre jug you can make 2, 4 and 6 litres, but never 1, 3 or 5, because every combination of 4s and 6s is even: HCF(4, 6) = 2.

Measuring 4 litres with 3 L and 5 L jugs

Step 01Fill the 5 L jug(0, 5)
Small jug empty, big jug 5 litres.
Step 02Pour big into small(3, 2)
The 3 L jug fills up; 2 litres stay in the big jug.
Step 03Empty the small jug(0, 2)
Pour the 3 litres away.
Step 04Pour big into small(2, 0)
The 2 litres move into the small jug.
Step 05Fill the 5 L jug(2, 5)
Small jug has 2 litres, big jug full.
Step 06Pour big into small(3, 4)
Only 1 litre fits into the small jug, so exactly 4 litres remain in the big jug.

Try it

Chapter 06

Ancient remainder puzzles: the Chinese Remainder Theorem

Around the 3rd to 5th century CE, the Chinese book Sunzi Suanjing posed this puzzle:

There are some things whose number is unknown. Counted in threes, 2 are left over; counted in fives, 3 are left over; counted in sevens, 2 are left over. How many things are there?

The smallest answer is 23. And here is where the LCM comes in: the answers repeat every LCM(3, 5, 7) = 105, giving 23, 128, 233, 338, …

The general result is called the Chinese Remainder Theorem: if the divisors are pairwise co-prime, then for any choice of remainders there is exactly one answer between 0 and the LCM, and the rest are that answer plus multiples of the LCM. Indian mathematicians, from Aryabhata’s kuttaka to Brahmagupta and Bhaskara, developed methods for the same kind of problem, often to work out when astronomical cycles would line up.

Worked example

0 / 4 steps shown

Solving a remainder puzzle by sieving

Find the smallest number that leaves remainder 1 when divided by 3 and remainder 2 when divided by 4.

Worked example

0 / 4 steps shown

The egg-basket puzzle

A woman carrying a basket of eggs is bumped and the eggs break. She cannot remember how many there were, but when she took them out 2, 3, 4, 5 or 6 at a time, one egg was always left over; taken out 7 at a time, none were left. What is the smallest number of eggs she could have had?

Try it

Chapter 07

Screens, ratios and HCF of fractions

Every screen size is a ratio in disguise. A Full HD screen is 1920 × 1080 pixels. HCF(1920, 1080) = 120, and 1920 ÷ 120 = 16 while 1080 ÷ 120 = 9, so its shape is 16 : 9. The screen is exactly a 16 × 9 grid of 120-pixel squares, the biggest square "tile" that fits, just like the courtyard in Discover.

TableScreen resolutions simplified with the HCF (all computed)
Resolution	HCF	Aspect ratio	Where you meet it
1920 × 1080	120	16 : 9	Full HD TVs, laptops
1280 × 720	80	16 : 9	HD video
3840 × 2160	240	16 : 9	4K TVs
2400 × 1080	120	20 : 9	many phones held sideways
1024 × 768	256	4 : 3	older tablets and monitors
2560 × 1600	320	8 : 5	some laptops

Lab

Factorise screen dimensions and use the shared primes to find each screen’s aspect ratio.

Round 1 / 7★ 0 ptsBest: 0

Key: double green ring = prime leaf · dashed = still to do · thick amber ring = the branch you’re working on.

Choose two numbers that multiply to make 1920.

Split 1920 into a factor pair:

Wrong tries on this tree: 0

Text version of this activity

This lab builds factor trees for screen widths and heights.

1,920 = 2⁷ × 3 × 5
1,080 = 2³ × 3³ × 5
1,280 = 2⁸ × 5
720 = 2⁴ × 3² × 5
2,400 = 2⁵ × 3 × 5²
3,840 = 2⁸ × 3 × 5
2,160 = 2⁴ × 3³ × 5

Shared part of 1,920 and 1,080: 2³ × 3 × 5 = 120, leaving 2⁴ = 16 and 3² = 9: the ratio 16 : 9. For 2,400 and 1,080 the shared part is also 120, leaving 20 and 9: the ratio 20 : 9. 3,840 × 2,160 is exactly twice 1,920 × 1,080, so the ratio stays 16 : 9.

Can fractions have an HCF and LCM? Yes, if you ask the right question. Two runners take ¾ minute and ⅚ minute per lap. When are they next together at the start? We need the smallest time that is a whole number of laps for both: a common multiple of ¾ and ⅚.

The rule is: LCM of fractions = LCM of the numerators ÷ HCF of the denominators (with each fraction in simplest form). Here LCM(3, 5) ÷ HCF(4, 6) = 15 ÷ 2 = 7½ minutes. Check: 7½ ÷ ¾ = 10 laps and 7½ ÷ ⅚ = 9 laps, both whole. ✓

Similarly, HCF of fractions = HCF of the numerators ÷ LCM of the denominators: the largest length that fits a whole number of times into both ¾ m and ⅚ m is HCF(3, 5) ÷ LCM(4, 6) = 1⁄12 m, which fits 9 and 10 times.

Try it

Chapter 08

Olympiad corner

These problems use nothing beyond this topic, but they need you to combine ideas. Try each for ten minutes before opening the solution.

Worked example

0 / 5 steps shown

Sum and HCF

Two numbers add up to 528 and their HCF is 33. How many such pairs are there?

Worked example

0 / 5 steps shown

Divisible by 4 or 6

How many numbers from 1 to 1,000 are divisible by 4 or 6 (or both)?

Worked example

0 / 5 steps shown

The smallest number every digit divides

What is the smallest positive number that is divisible by every whole number from 1 to 10? And from 1 to 12?

Predict first

LCM(1, 2, …, n) is 2,520 for n = 10. For which of these n does the LCM stay the same when you go from n − 1 to n?

n = 11n = 13n = 14n = 16

Try it

Lab

Classify real-world and puzzle problems by whether they need the HCF, the LCM, or both.

What does each problem need: the HCF, the LCM, or both?

12 cards, 3 bins. Tap a card, then tap its bin. You can also drag, or press a bin’s number key.

Text version of this activity

This game has twelve cards and three bins: HCF, LCM and Both.

HCF: simplifying 1920 × 1080 to 16 : 9 (HCF 120); which amounts 4 L and 6 L jugs can measure (multiples of 2); simplifying the 48 : 16 gear ratio to 3 : 1; squares crossed by the diagonal of a 12 × 18 grid (12 + 18 − 6 = 24).

LCM: Teentaal and Jhaptaal landing on sam together (80 beats); Sunzi’s remainder puzzle (answers repeat every 105; smallest 23); when 12- and 18-tooth gears repeat (36 teeth); numbers up to 1,000 divisible by 4 or 6 (333, subtracting multiples of 12); a 17-year cicada and a 6-year predator (every 102 years).

Both: pairs with HCF 12 and LCM 144 (split 12 into co-prime parts); whether offset 4- and 6-year cycles can meet (HCF decides if, LCM decides how often); runners with ¾ and ⅚ minute laps (LCM of numerators ÷ HCF of denominators = 7½ minutes).

Chapter 09

More puzzles from daily life in India

Worked example

0 / 4 steps shown

Temple bells from morning to evening

Three temple bells toll at intervals of 9, 12 and 15 minutes. They toll together at 9:00 a.m. How many times do they toll together from 9:00 a.m. up to and including 6:00 p.m.?

Worked example

0 / 3 steps shown

Two trains at a platform

At a junction, a Vande Bharat train passes every 25 minutes and a goods train every 40 minutes (in this made-up timetable). Both pass at noon. When do they next pass at the same time?

Worked example

0 / 4 steps shown

A rangoli on a dot grid

Priya’s rangoli board is 84 cm × 60 cm. She wants to mark dots in a square grid so that there are dots on all four edges and the corners, with the dots as far apart as possible. How far apart should they be, and how many dots will there be?

Try it

Worked example

0 / 4 steps shown

HCF of fractions: cutting ribbons

A craft teacher has ribbons of 2½ m and 3¾ m. She wants to cut both into equal pieces, as long as possible, with nothing left. How long is each piece?

Predict first

Which is bigger: LCM(1, 2, …, 20) ÷ LCM(1, 2, …, 19), or LCM(1, 2, …, 19) ÷ LCM(1, 2, …, 18)?

The firstThe secondThey are equal

Try it

Chapter 10

Who uses HCF and LCM?

Explore

HCF and LCM at work

Pick a job to see where the ideas turn up.

Fractions in code
→
Simplify with gcd
→
Euclid in a loop
→
Fast for huge numbers

Uses Euclid daily

Programming languages such as Python include gcd and lcm functions, built on Euclid’s algorithm. They are used to keep fractions in simplest form, to resize images without distortion, and to schedule repeating tasks. Every time your phone shows a photo in a 4 : 3 or 16 : 9 frame, a gcd was probably computed.

Chapter 11

Projects and open questions

Try it

Project ideas

School timetable audit. Find two things at school that repeat on different cycles (bells, sports days, library periods). Work out when they coincide over a term, then check against the real calendar.
Gear lab. Using LEGO or cardboard gears, mark one tooth on each of two meshing gears. Count turns until the marks meet again. Compare with the LCM. Then change one gear by one tooth.
Rhythm composer. Write a short piece for two clappers in cycles of 3 and 4 (or 5 and 7). Mark where they land together, and perform it.
Screen survey. Collect the resolutions of phones, laptops and TVs at home. Use the HCF to find each aspect ratio and make a table. Which ratio is most common?
Euclid speed test. Write down 10 pairs of numbers and count the division steps for each. Can you find a pair under 1,000 that beats the Fibonacci pair (987, 610)?

Think about it

Open questions for curious learners

Nobody has fully solved these yet.

The lonely runner conjecture. Runners with different constant speeds start together on a circular track. Is every runner, at some moment, far from all the others (at least 1⁄(number of runners) of a lap away)? It has been proved for up to 13 runners — by hand up to 7, and for 8 to 13 only recently, with computer-assisted searches — but nobody has proved it for every number of runners.
Why prime cycles? Do periodical cicadas really have 13- and 17-year cycles because of predators, because prime cycles keep broods from interbreeding, or for some other reason? Biologists still debate it.
Fast factorising. Euclid finds an HCF in a blink, but nobody knows a fast way to factorise huge numbers on ordinary computers. If someone found one, much of today’s internet security would need to change.
Fairer schedules. How should a city choose bus and metro frequencies so that connections are frequent but no route waits too long? This is an active research area mixing LCMs with data.

Chapter 12

Check yourself

Lab

Match real-world cycles and puzzles from this layer to their answers.

Match each situation to its answer.

16 face-down cards hide 8 pairs. Flip two at a time and remember where things are!

Text version of this activity

This memory game has sixteen cards forming eight pairs:

1920 × 1080 screen → 16 : 9 (divide by HCF 120).
Teentaal (16) and Jhaptaal (10) → sams meet every 80 beats.
Sunzi’s puzzle → 23.
Egg basket → 301 eggs.
LCM of 1 to 10 → 2,520.
Gears of 12 and 17 teeth → the same teeth meet again after 204 teeth.
Runners with ¾ and ⅚ minute laps → together after 7½ minutes.
Olympics (years divisible by 4) and FIFA World Cup (2 years later) → never in the same year.

Keep this

Cheat sheet

Cycles with a head start can meet only if the difference in starts is a multiple of the HCF; then they meet every LCM.
Real sky cycles are not whole numbers: alignments come from relative motion (Jupiter–Saturn ≈ 19.9 years). Calendars use near-LCMs like the 19-year Metonic cycle.
Cicadas with 13- and 17-year cycles meet short predator cycles rarely (big LCMs): a debated hypothesis.
Gears: the same teeth meet again after LCM(teeth) teeth; co-prime counts (a hunting tooth) spread wear evenly. Gear ratios are simplified with the HCF.
Rhythm: tala cycles land on sam together every LCM of their beats: Teentaal and Jhaptaal every 80.
Jugs: two jugs can measure exactly the multiples of their HCF (up to the bigger jug).
Remainder puzzles: solutions repeat every LCM of the divisors (Chinese Remainder Theorem).
Screens: divide width and height by their HCF to get the aspect ratio: 1920 × 1080 → 16 : 9.
Fractions: LCM = LCM(numerators) ÷ HCF(denominators); HCF = HCF(numerators) ÷ LCM(denominators).
Counting "a or b": multiples of a + multiples of b − multiples of LCM(a, b).

Reflect

Choose one expedition from this layer (calendars, cicadas, gears, rhythm, jugs, remainders or screens). Explain it to someone at home using your own numbers, and write down one question it raised that you could not answer.

This stays on this page only. It isn’t saved or sent anywhere.

Related to

Number and shape patterns

Cycles, rhythms and repeating remainders are patterns whose period is an LCM.

Used in

Data handling

Screen-size surveys and timetable audits turn HCF and LCM into real data projects.

Used in

Shape and space

Tiling rectangles, grid diagonals and gear circles bring HCF and LCM into geometry.

Related to

Electricity

Blinking lights, generator poles and AC cycles are repeating signals whose coincidences follow the LCM.

Where this comes from

Sources

Least common multiple (opens another website) — Wikipediaawaiting owner check
Reference for properties of the LCM: common multiples of two numbers are the multiples of their LCM, the prime-power (Venn) method, the recursive identity for three or more numbers, and the gear and planetary-alignment examples.
Greatest common divisor (opens another website) — Wikipediaawaiting owner check
Reference for properties of the GCD: every common divisor divides the GCD, gcd(a, 0) = |a|, co-prime numbers, the gcd × lcm identity for two numbers and Bézout’s identity.
Gear train (opens another website) — Wikipediaawaiting owner check
Supports gear ratios from tooth counts and the section on hunting and non-hunting gear sets: when tooth counts are relatively prime, every tooth meets every tooth of the other gear, giving less wear and longer life.
Chinese remainder theorem (opens another website) — Wikipediaawaiting owner check
Supports Sunzi’s remainder problem (remainders 2, 3, 2 on division by 3, 5, 7 gives 23) from the 3rd-to-5th-century Sunzi Suanjing, solutions repeating modulo the product of pairwise co-prime divisors, and work by Aryabhata (6th c.) and Brahmagupta (7th c.).
Euclidean algorithm (opens another website) — Wikipediaawaiting owner check
Supports the history (Euclid’s Elements, c. 300 BC, Book VII; Aryabhata’s late-5th-century 'pulveriser'; Qin Jiushao 1247), why the division method works, the original subtraction form, and the Fibonacci worst case (Lamé, 1844).
Least Common Multiple (opens another website) — Math is Funawaiting owner check
Supports the definition of multiple, common multiple and least common multiple, and finding the LCM of two or three numbers by listing multiples until the first match. (This page does not cover the prime-factorisation method or fractions.)
Metonic cycle (opens another website) — Wikipediaawaiting owner check
Supports the Metonic near-coincidence: 19 tropical years = 6,939.602 days and 235 synodic months = 6,939.689 days, differing by about 2 hours 5 minutes, and lunisolar calendars (Babylonian, Hebrew) adding 7 leap months in each 19-year cycle.
Hindu calendar (opens another website) — Wikipediaawaiting owner check
Supports the statement that Hindu lunisolar calendars insert an extra full month (adhik maas) about once every 32-33 months so that festivals stay in the right season.
Great conjunction (opens another website) — Wikipediaawaiting owner check
Supports the Jupiter–Saturn great conjunction figures: a mean interval of about 19.859 Julian years, the conjunction of 21 December 2020 and the next one in 2040 — a relative-motion period, not an LCM.
Periodical cicadas (opens another website) — Wikipediaawaiting owner check
Supports the 13- and 17-year life cycles and their ranges in eastern North America, and the competing explanations offered for the prime-number cycles (predator avoidance and avoiding hybridisation between broods), which remain hypotheses rather than settled results.
Lonely runner conjecture (opens another website) — Wikipediaawaiting owner check
Supports the statement of the conjecture (with n runners each is at some moment at least 1/n of a lap from all the others) and the record of cases settled so far: n up to 7 by elementary arguments, and n = 8 to 13 by computer-assisted work in 2025-2026.
Tala (music) (opens another website) — Wikipediaawaiting owner check
Supports the Hindustani tala cycle lengths used in the rhythm chapter — Teentaal 16 beats, Jhaptaal 10, Ektaal 12, Rupak 7 — and the meaning of sam, the first and most emphasised beat of a cycle, where phrases resolve.

End of Extend

What you just read

Decide whether two cycles with different starting points can ever coincide, using the HCF.
Apply HCF and LCM to gears, rhythms, screens, calendars and jug puzzles, and explain where the models break down.
Solve remainder puzzles, including Sunzi’s, using the LCM and systematic listing.
Find the HCF and LCM of fractions and use them in problems.
Tackle olympiad-style problems that combine HCF, LCM and counting.

The web

Explore a connection

Builds on
Prime and composite numbers
Prime factorisation is the fastest route to both the HCF and the LCM.
Used in
Number and shape patterns
Two repeating cycles line up again after their LCM — the pattern behind blinking lights and bus timetables.
Used in
Shape and space
The largest square tile that fits a rectangular floor exactly has a side equal to the HCF of its length and width.

Want to save topics or ask for new ones? Invited families can connect a learning device. Everything here stays free to read without signing in.

Revision 1 · release preview-7e1cbbcc4f · accepted 20/09/2026

Words from the expeditions

Two festivals with a head start

The bicycle chain

Measuring 4 litres with 3 L and 5 L jugs

Solving a remainder puzzle by sieving

The egg-basket puzzle

Sum and HCF

Divisible by 4 or 6

The smallest number every digit divides

Temple bells from morning to evening

Two trains at a platform

A rangoli on a dot grid

HCF of fractions: cutting ribbons

HCF and LCM at work

Extend check

Cheat sheet

Sources

What you just read

Explore a connection