Find the greatest common factor (highest common factor) and least common multiple of two or three whole numbers, calculated exactly.
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With first number 24, second number 36, third number (optional) 0, the greatest common factor (gcf) is 12.
| First number | 24 |
|---|---|
| Second number | 36 |
| Third number (optional) | 0 |
| Greatest common factor (GCF) | 12 |
|---|---|
| Least common multiple (LCM) | 72 |
| Relationship | Not coprime: they share a factor greater than 1 |
The greatest common factor (also called the highest common factor or greatest common divisor) is the largest whole number that divides all the numbers exactly. The least common multiple is the smallest number that all of them divide into. Euclid's algorithm finds the GCF quickly even for very large numbers, and all arithmetic is done with exact integers.
GCF by Euclid's algorithm: gcd(a, b) = gcd(b, a mod b) until the remainder is 0. LCM(a, b) = a ÷ GCF(a, b) × b. The greatest common factor (also called the highest common factor or greatest common divisor) is the largest whole number that divides all the numbers exactly. The least common multiple is the smallest number that all of them divide into. Euclid's algorithm finds the GCF quickly even for very large numbers, and all arithmetic is done with exact integers.
With first number 24, second number 36, third number (optional) 0, the greatest common factor (gcf) is 12. Least common multiple (LCM): 72; Relationship: Not coprime: they share a factor greater than 1.
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12. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24; the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The largest one they share is 12. Their LCM is 72.
For two numbers, GCF × LCM = a × b. With 24 and 36, 12 × 72 = 864 = 24 × 36. That is why the LCM can be found by dividing the product by the GCF.
To add fractions with different denominators (the LCM is the lowest common denominator) and to find when repeating events line up, such as buses every 12 and 18 minutes meeting every 36 minutes.