GCD / LCM Calculator
- Start with the numbers: 48, 72.
- Euclidean algorithm for 48 and 72: 72 = 1 × 48 + 24; 48 = 2 × 24 + 0. The last divisor, 24, is their GCD.
- Prime factors: 48 = 2⁴ × 3; 72 = 2³ × 3².
- GCD uses the lowest power of each shared prime: 2³ × 3 = 24. LCM uses the highest power of every prime: 2⁴ × 3² = 144.
- Check with the link between them: 48 × 72 ÷ 24 = 144.
- Euclidean algorithm: divide the larger number by the smaller, then repeat with the divisor and the remainder until the remainder is 0. The last divisor is the GCD.
- Prime factors: the GCD takes the lowest power of each shared prime, and the LCM takes the highest power of every prime.
- GCD and LCM link: for two numbers, GCD × LCM = the first number × the second.
- 24 divides each number evenly: 48 ÷ 24 = 2, 72 ÷ 24 = 3.
- 144 is a multiple of each number: 144 ÷ 48 = 3, 144 ÷ 72 = 2.
- After dividing by the GCD, the numbers 2, 3 have no common factor left.
GCD / LCM Calculator: Greatest Common Divisor and Least Common Multiple
This free GCD and LCM calculator finds the greatest common divisor and the least common multiple of two or more whole numbers at the same time. It shows the Euclidean algorithm step by step, writes out the prime factors, and builds a table that makes it clear where each answer comes from. It is handy for simplifying fractions, finding common denominators and checking homework.
Quick answer: the GCD of 48 and 72 is 24 and their LCM is 144.
What Are the GCD and LCM?
The greatest common divisor, also called the greatest common factor or highest common factor, is the largest number that divides every number in your list without a remainder. The least common multiple, also called the lowest common multiple, is the smallest positive number that every number in the list divides into. One looks downwards for the biggest shared piece, and the other looks upwards for the first shared multiple.
How to Find the GCD with the Euclidean Algorithm
Divide the larger number by the smaller and note the remainder. Then divide the smaller number by that remainder, and keep going until the remainder is 0. The last divisor you used is the GCD. For 48 and 72, 72 = 1 × 48 + 24 and 48 = 2 × 24 + 0, so the GCD is 24. This method is fast even for very large numbers, which is why the calculator accepts values up to twelve digits.
Finding Both with Prime Factors
Write each number as a product of primes. The GCD takes the lowest power of every prime the numbers share, and the LCM takes the highest power of every prime that appears in any of them. The table in the results shows these powers in a grid, with the GCD and LCM rows highlighted. If you need the factors of a single number, the prime factorization calculator shows them with a factor tree. Some common pairs are shown below.
| Numbers | GCD | LCM |
|---|---|---|
| 12 and 18 | 6 | 36 |
| 48 and 72 | 24 | 144 |
| 8 and 15 | 1 | 120 |
| 100 and 75 | 25 | 300 |
| 14 and 49 | 7 | 98 |
| 12, 18 and 30 | 6 | 180 |
| 84, 126 and 210 | 42 | 1,260 |
The Link Between GCD and LCM
For two numbers, the GCD multiplied by the LCM equals the product of the numbers. This means the LCM is the product divided by the GCD, which is the quickest way to find it once the GCD is known. If the GCD is 1, the numbers are coprime, and their LCM is simply their product. With three or more numbers, the calculator works through them one pair at a time.
Where GCD and LCM Are Used
The GCD simplifies fractions: divide the top and bottom by it and the fraction is in lowest terms. The LCM gives the least common denominator, so it is used whenever fractions with different denominators are added. Both also come up in scheduling, such as finding when two repeating events line up, and in dividing items into equal groups. For the fraction work itself, try the fraction calculator, and for general arithmetic, the scientific calculator, the math solver and the percentage calculator.
How to Use This GCD / LCM Calculator
Enter your numbers
Type whole numbers in the boxes or drag the sliders. Two boxes are ready for you.
Add more if needed
Press Add another number to include up to eight numbers, and Remove to take one away.
Read the answer
See the GCD and the LCM, and whether the numbers are coprime.
Follow the working
Check the Euclidean steps, the prime factors and the table.