What is the LCM? Definition, Full Form, Examples
What is the LCM? The least common multiple (LCM) of two or more whole numbers is the smallest positive number that all of those numbers divide into exactly. LCM(4, 6) = 12, because 12 is the first number that appears in the multiples of both 4 and 6.
The LCM is also called the lowest common multiple in the United Kingdom and India. When the numbers are the denominators of fractions, the LCM is called the lowest common denominator (LCD). The LCM Calculator finds the LCM of up to 15 numbers and shows the working.
What is the LCM full form and notation?
LCM stands for Least Common Multiple (United States) or Lowest Common Multiple(United Kingdom, India). Mathematicians write the LCM of a and b as lcm(a, b), LCM(a, b) or [a, b]. The LCM of 4 and 6 is written LCM(4, 6) = 12.
Multiples and common multiples
A multiple of a number is that number times any whole number. The multiples of 6 are 6, 12, 18, 24, 30, 36, and so on. A common multiple of two numbers is a number in both lists of multiples. 12, 24 and 36 are common multiples of 4 and 6. The least common multiple is the smallest one: 12.
Worked example: LCM of 4 and 6
Listing multiples: LCM(4, 6)
Write out the multiples of each number until one multiple appears in every row. The first number that appears in all rows is the least common multiple.
12 is the first multiple shared by every number.
LCM(4, 6) = 12
The same answer comes from prime factors: 4 = 2² and 6 = 2 × 3, so the LCM = 2² × 3 = 12. The four standard methods are on How to find the LCM.
Multiples and LCM of a single number
A single number has no least common multiple on its own. "LCM of 36" usually means one of three things: the multiples of 36 (36, 72, 108, …), the LCM of 36 with a second number that was left out, or the prime factorization 36 = 2² × 3². To find the LCM of 36 and another number, enter both in the calculator.
LCM of 1 to 10
The LCM of 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 is 2,520.
2,520 = 2³ × 3² × 5 × 7: the highest power of 2 up to 10 is 8, the highest power of 3 is 9, and 5 and 7 appear once each. 2,520 is the smallest number that every number from 1 to 10 divides.
| Range | LCM of 1 to n | Prime powers |
|---|---|---|
| 1 to 2 | 2 | 2 |
| 1 to 3 | 6 | 2 × 3 |
| 1 to 4 | 12 | 2² × 3 |
| 1 to 5 | 60 | 2² × 3 × 5 |
| 1 to 6 | 60 | 2² × 3 × 5 |
| 1 to 7 | 420 | 2² × 3 × 5 × 7 |
| 1 to 8 | 840 | 2³ × 3 × 5 × 7 |
| 1 to 9 | 2,520 | 2³ × 3² × 5 × 7 |
| 1 to 10 | 2,520 | 2³ × 3² × 5 × 7 |
What is the LCM of 1 to 10 except 7?
Without 7 the LCM drops to 360, because 7 was the only number contributing the factor 7. 2,520 ÷ 7 = 360.
Special cases: 0, 1 and negative numbers
- LCM with 0. By convention LCM(a, 0) = 0, and some textbooks call it undefined. The calculator does not accept 0.
- LCM with 1. LCM(1, n) = n. The number 1 changes nothing.
- Negative numbers. The LCM uses absolute values and is always positive: LCM(−4, 6) = 12.
- Equal numbers. LCM(n, n) = n.
- Coprime numbers. Numbers with no shared factor have LCM equal to their product: LCM(7, 11) = 77.
LCM and the greatest common factor
The greatest common factor (GCF, called HCF in India and the UK) is the largest divisor shared by the numbers. For two numbers, LCM × GCF = a × b. For 4 and 6: LCM 12 × GCF 2 = 24 = 4 × 6. The full comparison is on What is the LCM? Definition, full form, examples's sister page, LCM vs HCF.
Where the LCM is used
The LCM adds and subtracts fractions through the lowest common denominator, finds when repeating events line up (two buses every 12 and 15 minutes meet again after 60 minutes), and appears in the gears problem and in scheduling. Every number from 1 upward divides 2,520, which is why 2,520 appears in puzzle questions.
What grade or class teaches the LCM?
United States: grade 6, Common Core 6.NS.B.4 (GCF up to 100, LCM up to 12). India: introduced in class 4 and 5, formal in class 6 (NCERT), revisited in class 10 with the rule HCF × LCM = product of two numbers. United Kingdom: Key Stage 3 and GCSE number topics.
Frequently asked questions
What is the full form of LCM?
LCM stands for Least Common Multiple in the United States and Lowest Common Multiple in the United Kingdom and India. Both names describe the same number: the smallest positive multiple shared by two or more whole numbers.
What is an example of LCM?
LCM(4, 6) = 12. The multiples of 4 are 4, 8, 12, 16 and the multiples of 6 are 6, 12, 18. The first multiple in both lists is 12, so 12 is the least common multiple of 4 and 6.
What is the LCM of 1 to 5?
The LCM of 1, 2, 3, 4 and 5 is 60. 60 = 2² × 3 × 5, the highest power of each prime in the five numbers.
How do you find the LCM of 1 to 10?
The LCM of 1 to 10 is 2,520. Take the highest power of each prime up to 10: 2³ × 3² × 5 × 7 = 2,520. Every number from 1 to 10 divides 2,520 exactly.
Can the LCM be negative?
No. The least common multiple is always a positive whole number. Negative inputs use their absolute values, so LCM(−2, −3) = LCM(2, 3) = 6.
Can the LCM be 1?
Yes, only when every number is 1. LCM(1, 1) = 1. For any other numbers the LCM is at least as large as the largest number.
What is the LCM of 12 and 3?
The LCM of 12 and 3 is 12. When one number is a multiple of the other, the larger number is the least common multiple.
What is the least multiple of 6?
The least multiple of 6 is 6 itself, because 6 × 1 = 6. The multiples of 6 continue 12, 18, 24 and so on. The phrase "least common multiple" needs two or more numbers to compare.
Is there a formula for the LCM?
Yes, for two numbers: LCM(a, b) = a × b ÷ GCF(a, b). For 12 and 18 the GCF is 6, so the LCM = 12 × 18 ÷ 6 = 36. For three or more numbers, use prime factorization.
What grade is the LCM taught in?
In the United States the LCM is taught in grade 6 under Common Core standard 6.NS.B.4, with LCMs up to 12. In India the LCM is introduced in class 4 and 5 and taught formally in class 6 (NCERT, "Playing with Numbers").
Sources
- Wikipedia: Least common multiple: definition, notation, LCM × GCD = a × b
- Wolfram MathWorld: Least Common Multiple: formal properties
- Common Core 6.NS.B.4: GCF and LCM in grade 6