Where two counts first meet
Count in fours: 4, 8, 12, 16, 20, 24. Then count in sixes: 6, 12, 18, 24. The two counts first land on the same number at 12.
The smallest number that is a multiple of both of two numbers is called their Lowest Common Multiple, or LCM for short. The LCM of 4 and 6 is 12.
Counting in fours lands on 4, 8, 12, 16, 20 and 24.
On the same scale, counting in sixes lands on 6, 12, 18 and 24. 12 is the first landing the two counts share.
The first meeting, not a later one
24 is on both counts as well, but it comes later. The LCM is the lowest common multiple, so it is 12, not 24.
Every other common multiple of 4 and 6 is a multiple of the LCM: 24 = 12 × 2 and 36 = 12 × 3. After the first meeting, the two counts meet again every 12.
The LCM from the primes
Write both numbers as primes: 4 = 2 × 2 and 6 = 2 × 3. A multiple of 4 has 2 × 2 inside it, and a multiple of 6 has 2 × 3 inside it. The smallest number with both inside it is 2 × 2 × 3 = 12. Its two 2s are what 4 needs, and one of those 2s with the 3 is what 6 needs.
So take every prime that appears in either list, as many times as the list that has it most. 2 appears twice in 4 and once in 6, so take it twice. 3 appears only in 6, once, so take it once. 2 × 2 × 3 = 12.
4 = 2 × 2 and 6 = 2 × 3 share one 2, which goes in the overlap once. Everything in the diagram multiplied together, 2 × 2 × 3 = 12, is the LCM.
Bigger numbers
For 24 and 36: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. 2 appears three times in 24 and twice in 36, so take it three times. 3 appears once in 24 and twice in 36, so take it twice. The LCM is 2 × 2 × 2 × 3 × 3 = 72.
The primes the two lists share, 2 × 2 × 3 = 12, make the HCF. The HCF takes what both lists have; the LCM takes everything either list has.
0 of the 3 shared factors are in the overlap, product 1 so far; everything drawn, each shared factor once, multiplies to 2 × 2 × 2 × 3 × 3 = 72 whatever is in the middle
Slide the shared factors into the overlap
Slide the shared primes of 24 and 36 into the overlap, one at a time. The overlap multiplies to the HCF, 12. Everything in the diagram, with each shared prime counted once, multiplies to the LCM, 72.
A check
The HCF multiplied by the LCM always equals the two numbers multiplied. For 24 and 36: 12 × 72 = 864, and 24 × 36 = 864. For 4 and 6: the HCF is 2, and 2 × 12 = 24 = 4 × 6.
Multiplying can overshoot
Multiplying the two numbers gives a common multiple, but not always the lowest. 4 × 6 = 24 uses the shared 2 twice, once for 4 and once for 6, when the LCM needs it only once.
The product is the LCM only when the two numbers share no factor except 1. 8 and 9 share none, so their LCM is 8 × 9 = 72.
Adding is never the way: 4 + 6 = 10 is not a multiple of 4 or of 6. And the LCM is never smaller than the larger number, because the larger number divides it.
Worked example: LCM: Equal Quantity Matching Across Different Pack Sizes
Question At a bakery, hotdog buns are sold in packs of 6 for $4, and sausages are sold in packs of 8 for $5. Melissa needs to buy the exact same number of hotdog buns and sausages to make hotdogs, without having any buns or sausages left over. What is the least amount of money she can spend in total?
1.List bun counts: 6, 12, 18, 24, 30...
Buns come in sixes: 6, 12, 18, 24, 30. 2.List sausage counts: 8, 16, 24, 32...
Sausages come in eights: 8, 16, 24, 32. 3.First common quantity match is 24 items.
The first count both lists reach is 24, the LCM of 6 and 8. 4.Draw 4 unit packs for buns: 4 × $4 = $16.
24 buns is 4 packs: 4 × $4 = $16. 5.Draw 3 unit packs for sausages: 3 × $5 = $15.
24 sausages is 3 packs: 3 × $5 = $15. 6.Combine costs: $16 + $15 = $31.
Least spend: $16 + $15 = $31.
Answer: $31
Common mistakes
- Finding the LCM of the prices ( LCM(4, 5) = 20 ) instead of the pack quantities.
- Assuming that buying 1 pack of each yields equal items (6 ≠ 8).
- Multiplying 24 items directly by the pack price rather than dividing by pack size first.
Worked example: LCM: Periodic Synchronization (Simultaneous Events)
Question Three lighthouses flash their lights at regular intervals. Lighthouse A flashes every 24 seconds, Lighthouse B flashes every 36 seconds, and Lighthouse C flashes every 60 seconds. All three lighthouses flash simultaneously at 8:00 p.m. (a) At what time will they next flash together? (b) How many times will all three flash together between 8:01 p.m. and 8:35 p.m.?
1.Ladder method for LCM of (24, 36, 60):
Three rhythms on one clock. They coincide again at the first common multiple. 2.Divide by 12 → (2, 3, 5).
Ladder: all three divide by 12, leaving 2, 3, 5. 3.Since 2, 3, 5 are pairwise coprime, multiply: 12 × 2 × 3 × 5 = 360 seconds = 6 minutes.
Those share nothing, so multiply everything: 12 × 2 × 3 × 5 = 360 s, which is 6 minutes. 4.Step through intervals: 8:00, 8:06, 8:12, 8:18, 8:24, 8:30, 8:36 (exceeds 8:35).
(a) Next together at 8:06 p.m. Then every 6 minutes. 5.Count matches strictly between 8:01 and 8:35: 5 occurrences.
(b) Strictly between 8:01 and 8:35: 8:06, 8:12, 8:18, 8:24, 8:30. Five times; 8:36 is out.
Answer: (a) 8:06 p.m.; (b) 5 times
Common mistakes
- Selecting the HCF instead of LCM (giving HCF = 12 seconds, which is far too short to align with a 36- or 60-second cycle).
- Including the 8:00 p.m. event when the question specifies 'between 8:01 p.m. and 8:35 p.m.'.
- Forgetting to convert 360 seconds into minutes when stating the clock time.