Least Common Factor: Why It Is Always 1
The least common factor of any two or more whole numbers is 1, because 1 divides every whole number. A question about the "least common factor" usually means the least common multiple (LCM) or the greatest common factor (GCF).
Why Is the Least Common Factor Always 1?
Every whole number is divisible by 1, so 1 is on every list of factors and is always the smallest entry.
The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The numbers on both lists, the common factors, are 1, 2, 3, 6. The least of them is 1, and the same happens for any pair.
Least Common Factor, LCM or GCF: Which One Do You Need?
Ask what the answer is for: equal groups need the GCF, matching cycles need the LCM. The least common factor answers no practical question.
| You want to… | You need | 12 and 18 |
|---|---|---|
| Split two amounts into equal groups that are as large as possible | GCF (greatest common factor, also HCF) | 6 |
| Find when two repeating things line up again, or a common denominator | LCM (least common multiple) | 36 |
| Simplify a fraction | GCF | 12/18 = 2/3 |
| Add fractions with different denominators | LCM of the denominators | 36 |
| The smallest number dividing both | Least common factor | 1 (always) |
The difference between the two useful ones is explained in full on LCM vs HCF (GCF).
Least Common Factor of 12 and 18, Worked Both Ways
The same two numbers give three different answers depending on the question:
- Least common factor: common factors 1, 2, 3, 6 → 1.
- Greatest common factor: the largest common factor → 6. By primes: 12 = 2² × 3, 18 = 2 × 3²; take the lowest power of each shared prime.
- Least common multiple: take every prime at its highest power: 2² × 3² = 36. Check: 36 ÷ 12 = 3, 36 ÷ 18 = 2.
The step-by-step methods for the LCM are on How to Find the LCM.
The Smallest Common Factor Greater Than 1
The smallest common factor above 1 is the smallest prime that divides the GCF; when the GCF is 1 there is none.
| Numbers | GCF | Smallest common factor above 1 | LCM |
|---|---|---|---|
| 8 and 12 | 4 | 2 | 24 |
| 9 and 15 | 3 | 3 | 45 |
| 14 and 21 | 7 | 7 | 42 |
| 10 and 25 | 5 | 5 | 50 |
| 7 and 9 | 1 | none | 63 |
| 16 and 24 | 8 | 2 | 48 |
Numbers whose GCF is 1, such as 7 and 9, are called coprime: they share no factor except 1.
Where the Mix-Up Comes From
Two correct terms share words: least common multiple and greatest common factor. Swapping one word gives "least common factor", a phrase with a fixed answer of 1.
Least Multiple Calculator: the LCM and the GCF of 2 to 15 numbers, worked out six ways.
Frequently asked questions
What is the least common factor of two numbers?
The least common factor of any two whole numbers is 1, because 1 divides every number. The common factors of 12 and 18 are 1, 2, 3, 6, and the smallest is 1.
Is the least common factor the same as the LCM?
No. A factor divides a number; a multiple is a number times a whole number. The least common factor is always 1, while the LCM of 12 and 18 is 36.
What is the difference between the least common factor and the greatest common factor?
Both come from the same list of common factors. The least is always 1; the greatest is the GCF. For 12 and 18 the list is 1, 2, 3, 6, so the GCF is 6.
What is the smallest common factor greater than 1?
It is the smallest prime that divides the GCF. For 12 and 18 the GCF is 6, so the answer is 2. When the GCF is 1 there is none: 7 and 9 share only 1.
Why do people search for the least common factor?
Usually the question mixes up two terms: "least common multiple" and "greatest common factor". The words "least" and "factor" end up in one phrase.
Can the least common factor ever be bigger than 1?
No, not for whole numbers. 1 is a factor of every whole number, so it is always shared and always the smallest.
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