Least common multiple

Key takeaways

Least common multiple belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Least common multiple to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Least common multiple from memory before moving on to harder problems.

Reference excerpt

In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a, 0) as 0 for all a, since 0 is the only common multiple of a and 0. The least common multiple of the denominators of two fractions is the "lowest common denominator" (lcd), and can be used for adding, subtracting or comparing the fractions. The least common multiple of more than two integers a, b, c, . . . , usually denoted by lcm(a, b, c, . . .), is defined as the smallest positive integer that is divisible by each of a, b, c, . . .

Overview A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.

Notation The least common multiple of two integers a and b is denoted as lcm(a, b). Some older textbooks use [a, b].

Example

lcm ⁡ ( 4 , 6 ) {\displaystyle \operatorname {lcm} (4,6)}

Multiples of 4 are:

4 , 8 , 12 , 16 , 20 , 24 , 28 , 32 , 36 , 40 , 44 , 48 , 52 , 56 , 60 , 64 , 68 , 72 , 76 , . . . {\displaystyle 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,...}

Multiples of 6 are:

6 , 12 , 18 , 24 , 30 , 36 , 42 , 48 , 54 , 60 , 66 , 72 , . . . {\displaystyle 6,12,18,24,30,36,42,48,54,60,66,72,...}

Common multiples of 4 and 6 are the numbers that are in both lists:

12 , 24 , 36 , 48 , 60 , 72 , . . . {\displaystyle 12,24,36,48,60,72,...}

In this list, the smallest number is 12. Hence, the least common multiple is 12.

Applications When adding, subtracting, or comparing simple fractions, the least common multiple of the denominators (often called the lowest common denominator) is used, because each of the fractions can be expressed as a fraction with this denominator. For example,

2 21 + 1 6 = 4 42 + 7 42 = 11 42 {\displaystyle {2 \over 21}+{1 \over 6}={4 \over 42}+{7 \over 42}={11 \over 42}}

where the denominator 42 was used, because it is the least common multiple of 21 and 6.

Gears problem Suppose there are two meshing gears in a machine, having m and n teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete to realign the line segment can be calculated by using lcm ⁡ ( m , n ) {\displaystyle \operatorname {lcm} (m,n)} . The first gear must complete lcm ⁡ ( m , n ) m {\textstyle {\frac {\operatorname {lcm} (m,n)}{m}}} rotations for the realignment. By that time, the second gear will have made lcm ⁡ ( m , n ) n {\textstyle {\frac {\operatorname {lcm} (m,n)}{n}}} rotations.

Planetary alignment

Suppose there are three planets revolving around a star that take l, m, and n units of time, respectively, to complete their orbits. Assume that l, m, and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after lcm ⁡ ( l , m , n ) {\displaystyle \operatorname {lcm} (l,m,n)} units of time. At this time, the first, second and third planet will have completed lcm ⁡ ( l , m , n ) l {\textstyle {\frac {\operatorname {lcm} (l,m,n)}{l}}} , lcm ⁡ ( l , m , n ) m {\textstyle {\frac {\operatorname {lcm} (l,m,n)}{m}}} and lcm ⁡ ( l , m , n ) n {\textstyle {\frac {\operatorname {lcm} (l,m,n)}{n}}} orbits, respectively, around the star.

Calculation

There are several ways to compute least common multiples.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Least common multiple

Start with the simplest possible case. Write down what Least common multiple claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Least common multiple before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Least common multiple ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Least common multiple

In research: Least common multiple appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Least common multiple in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Least common multiple is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary arithmetic, Number theory, Operations on numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Least common multiple outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Least common multiple in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Least common multiple means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Least common multiple out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Least common multiple in simple terms?

In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, thi…

Why does Least common multiple matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Least common multiple?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Least common multiple.