Finding Common Ground · Lesson 4 of 7
Common Multiples and the LCM
“Find the first length or time shared by two repeating patterns.”
• Identify common multiples and define the least common multiple. • Recognise repeating-event and shared-length problems as LCM situations. • Build the LCM from the largest required count of each prime. • Check a result by dividing it by every given number.
The first shared length
Two makers join 6 cm and 8 cm strips, respectively, to make torans of equal length. The first maker can reach 6, 12, 18, 24, ... cm; the other reaches 8, 16, 24, ... cm. Their first shared length is 24 cm. It is a multiple of both strip lengths, and no smaller positive length works. Common multiples continue without bound, so there is no greatest common multiple.
The smallest positive number that is a multiple of every given positive whole number; also called the least common multiple.
Problem
Find the shortest equal length made using only 6 cm strips or only 8 cm strips.
- 1.Multiples of 6 begin 6, 12, 18, 24. Multiples of 8 begin 8, 16, 24.
- 2.Their first common multiple is 24.
- 3.Four 6 cm strips and three 8 cm strips each make 24 cm.
When repeating schedules coincide
A free sweet is offered every Monday, while a visitor returns every 10 days. Starting together on a Monday, future Mondays occur after multiples of 7 days, and future visits after multiples of 10 days. The first day in both lists is 70. The Idli-Vada counting game uses the same first-common-multiple idea when both calls occur together.
Problem
Starting together today, when do a seven-day and a ten-day cycle next coincide?
- 1.List or compare multiples: 7, 14, ... and 10, 20, ... .
- 2.The first common multiple is 70.
- 3.It represents ten weekly cycles and seven ten-day cycles.
Build the smallest number containing both factorisations
A multiple must contain all the prime pieces of the number it is a multiple of. To be divisible by two numbers at once, include enough copies of each prime for both. Taking more copies makes a larger common multiple; taking fewer fails for at least one number. Therefore, use the larger occurrence count for every prime.
Compare 36 = 2 × 2 × 3 × 3 with its multiple 648 = 36 × 18. The factorisation of 648 contains all four prime pieces of 36 and the additional pieces from 18. This subproduct relationship explains how prime pieces identify a multiple.
For 14 = 2 × 7 and 35 = 5 × 7, include one 2, one 5 and one shared 7: 70. Multiplying 14 × 35 would include 7 twice and gives a common multiple, but not the lowest one.
Problem
Find the LCM of 96 and 360.
- 1.96 = 2⁵ × 3; 360 = 2³ × 3² × 5.
- 2.Take five 2s, two 3s and one 5, the largest counts needed.
- 3.LCM = 2⁵ × 3² × 5 = 1440. Both 96 and 360 divide 1440.
Problem
Find the LCM of 6, 8 and 9.
- 1.6 = 2 × 3; 8 = 2³; 9 = 3².
- 2.The largest counts are three 2s and two 3s.
- 3.LCM = 2³ × 3² = 72; all three numbers divide it.
HCF keeps only shared primes and takes the fewest copies. LCM must accommodate all given numbers and takes the most copies of every prime appearing anywhere.
Quiz
What is the LCM of 6 and 8?
Which prime count belongs in the LCM of 2² × 3 and 2⁴ × 5?
What is the first common positive multiple of 7 and 10?
Does a greatest common multiple of 6 and 8 exist?
Which is the LCM of 14 and 35?
Practice Problems
- Find the LCM of 30 and 72 by prime factorisation.
- Find the LCM of 36 and 54, showing how many 2s and 3s are needed.
- Two signals flash every 14 and 30 seconds, starting together. When do they next flash together?
- Find the LCM of 105, 195 and 65.
- Explain why 14 × 35 is a common multiple but is not the least one.
- Find the LCM of 222 and 370 and verify both divisions.
Key Takeaways
• A common multiple is divisible by every given number. • LCM is the smallest positive common multiple. • First shared length and first repeat-together time are common LCM settings. • For each prime, the LCM needs the largest count appearing in any factorisation. • Check that every original number divides the proposed LCM.