GCD
12
LCM
144
Coprime
No
Find the GCD and LCM of three or more numbers at once.
12
144
Verification
48 × 36 = 1728
GCD × LCM = 12 × 144 = 1728
GCD(a, b) = GCD(b, a mod b), base case: GCD(a, 0) = aRepeatedly divide and take remainders until the remainder is 0. The last non-zero remainder is the GCD.
Where:
a= First number (larger)b= Second number (smaller)mod= Modulo (remainder after division)LCM(a, b) = |a × b| / GCD(a, b)The LCM equals the absolute product of the two numbers divided by their GCD.
Where:
a= First numberb= Second numberGCD(a,b)= Greatest common divisor of a and bInputs
Result
Using the Euclidean algorithm: 48 = 1 × 36 + 12, then 36 = 3 × 12 + 0. GCD = 12. LCM = (48 × 36) / 12 = 1728 / 12 = 144.
Inputs
Result
100 = 1 × 75 + 25, then 75 = 3 × 25 + 0. GCD = 25. LCM = (100 × 75) / 25 = 7500 / 25 = 300.
Inputs
Result
17 = 1 × 13 + 4, 13 = 3 × 4 + 1, 4 = 4 × 1 + 0. GCD = 1 (coprime). LCM = 17 × 13 = 221.
The GCD of two or more numbers is the largest positive integer that divides each number without a remainder. For example, GCD(48, 36) = 12 because 12 is the largest number that evenly divides both 48 and 36.
| Number Pair | GCD | Method |
|---|---|---|
| 48, 36 | 12 | Both divisible by 12 |
| 100, 75 | 25 | Both divisible by 25 |
| 17, 13 | 1 | Coprime (no shared factors) |
| 84, 56 | 28 | Both divisible by 28 |
The LCM of two or more numbers is the smallest positive integer that is divisible by each number. For example, LCM(4, 6) = 12 because 12 is the smallest number divisible by both 4 and 6.
| Number Pair | LCM | Common Use |
|---|---|---|
| 4, 6 | 12 | Adding fractions 1/4 + 1/6 |
| 3, 5 | 15 | Scheduling every 3 and 5 days |
| 12, 8 | 24 | Finding common denominators |
| 15, 20 | 60 | Time calculations (minutes) |
The Euclidean algorithm finds the GCD by repeatedly dividing the larger number by the smaller and taking the remainder. When the remainder reaches 0, the last non-zero remainder is the GCD. For 48 and 36: 48 = 1 x 36 + 12, then 36 = 3 x 12 + 0, so GCD = 12.
| Step | Division | Remainder |
|---|---|---|
| 1 | 48 = 1 x 36 + 12 | 12 |
| 2 | 36 = 3 x 12 + 0 | 0 (done) |
| Result | GCD = 12 |
For any two positive integers a and b, the product of their GCD and LCM equals the product of the numbers themselves: GCD(a,b) x LCM(a,b) = a x b. This means LCM(a,b) = (a x b) / GCD(a,b), which is the most efficient way to compute LCM.
| Numbers | GCD | LCM | Product Check |
|---|---|---|---|
| 48, 36 | 12 | 144 | 12 x 144 = 1,728 = 48 x 36 |
| 15, 20 | 5 | 60 | 5 x 60 = 300 = 15 x 20 |
| 7, 11 | 1 | 77 | 1 x 77 = 77 = 7 x 11 |
GCD is used to simplify fractions (e.g., 48/36 becomes 4/3 by dividing by GCD 12), tile floors with the largest square tile, and distribute items equally. LCM is used to find common denominators, synchronize schedules, and solve gear/pulley ratio problems.
| Application | Uses GCD or LCM | Example |
|---|---|---|
| Simplify fractions | GCD | 48/36 = 4/3 |
| Common denominators | LCM | 1/4 + 1/6 = 5/12 |
| Floor tiling | GCD | 48x36 room: 12x12 tiles |
| Schedule sync | LCM | Buses at 15 & 20 min: every 60 min |
Last Updated: Mar 9, 2026
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