Least Common Multiple (LCM)
Numbers · Least Common Multiple
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 5 Least Common Multiple (LCM)
Lesson objectives
By the end of this topic, you should be able to:
- List multiples of a number.
- Find the LCM of a set of numbers by listing and by prime factorisation.
- Apply the LCM to real life situations.
- Use the LCM to add and subtract fractions with different denominators.
Least Common Multiple (LCM)
The LCM of two or more numbers is the smallest number that all of them divide into exactly.
a) Multiples
A multiple of a number is what you get by multiplying it by a whole number.
The multiples of 12 are 12, 24, 36, 48 and so on. The list never ends.
Every number has infinitely many multiples, but only a finite number of factors. That is the difference between the two words.
b) By listing multiples
Write out the multiples of each number.
Find the first one that appears in every list.
That is the LCM.
This works well for small numbers and becomes slow for large ones.
c) By prime factorisation
Write each number as a product of primes in index form.
Take every prime that appears in any of them.
Take each at its highest power.
Multiply those together.
d) Why the highest power
The LCM must be divisible by both numbers.
Since 12 contains 2², the LCM must contain at least 2² for 12 to divide into it.
The higher power satisfies both requirements at once. That is the reason for the rule.
Notice this is the exact opposite of the GCD rule, where the lowest power was used. Mixing the two is the commonest error in this topic.
e) The useful check
For any two numbers, GCD × LCM = the product of the numbers.
For 12 and 18: the GCD is 6 and the LCM is 36. Then 6 × 36 = 216, and 12 × 18 = 216. They match, so both answers are right.
This check takes seconds and catches almost every mistake.
f) Adding fractions
The LCM is what you need when adding fractions with different denominators.
To add 5/12 and 1/18, the common denominator is the LCM of 12 and 18, which is 36.
Then 5/12 becomes 15/36, and 1/18 becomes 2/36. The sum is 17/36.
Any common denominator works. The LCM keeps the numbers smallest, and often removes the need to simplify afterwards.
g) A real example
One bus leaves every 12 minutes and another every 18 minutes. They leave together at 8.00 a.m. When do they next leave together?
The answer is the LCM, 36 minutes. So at 8.36 a.m.
h) Where this is used
Adding and subtracting fractions. Working out when repeating events coincide. Scheduling. Gear and pulley problems in physics.
Words to know
- Multiple -- the result of multiplying a number by a whole number.
- Common multiple -- a number that is a multiple of two or more given numbers.
- LCM -- the smallest common multiple of a set of numbers.
- Lowest common denominator -- the LCM of the denominators of two or more fractions.
- Highest power -- the largest index of a prime across the numbers considered.
:::checkpoint Check yourself
- Find the LCM of 8 and 12 by listing multiples.
- Find the LCM of 24 and 36 by prime factorisation.
- Check your answer using GCD × LCM.
- Why do you take the highest power for the LCM but the lowest for the GCD? :::
Bridge to practice
The exercises begin with multiples and the listing method, move through prime factorisation and the contrast with GCD, and finish with word problems, fractions and the size checks. After every LCM answer, divide it by each original number to confirm the division is exact.