LCM & GCD Calculator
Find the Least Common Multiple and Greatest Common Divisor instantly
LCM & GCD Calculator
Find the Least Common Multiple and Greatest Common Divisor
Enter two positive integers
LCM(a,b) = (a x b) / GCD(a,b)What is an LCM & GCD Calculator?
An LCM & GCD Calculator is a math tool that finds two important number relationships:
- GCD (Greatest Common Divisor) -- the largest positive integer that divides each number with no remainder
- LCM (Least Common Multiple) -- the smallest positive integer that is a multiple of each number
These concepts are used constantly in arithmetic, fractions, and algebra. GCD is most commonly used to simplify fractions and reduce ratios. LCM is most commonly used to find common denominators, combine fractions, and solve problems involving repeating schedules or cycles.
Because finding LCM and GCD by hand can be slow—especially for large numbers or multiple values—this calculator lets you enter your numbers and get the correct result instantly.
How to Use This LCM & GCD Calculator
- Enter your numbers -- input two positive integers into the fields above
- Click 'Calculate' -- to compute both GCD and LCM
- Review both results -- the calculator displays the GCD and LCM simultaneously
- Use the output -- apply the GCD for fraction simplification or the LCM for common denominators, scheduling, or math problems
Tips:
- Use whole positive numbers for the most standard interpretation of LCM and GCD
- If you include 0, the behavior may differ depending on the definition used (many tools define gcd(a, 0) = |a| and lcm(a, 0) = 0)
- For multiple numbers, calculators typically compute results by combining them step-by-step (pairwise)
Formulas
Greatest Common Divisor (GCD)
The GCD of two numbers a and b is the largest integer that divides both without remainder. A common method is the Euclidean Algorithm:
- Divide: a = bq + r
- Replace: a ← b, b ← r
- Repeat until r = 0
The last non-zero remainder is the GCD. In short:
gcd(a, b) = gcd(b, a mod b)
until the remainder is 0
Least Common Multiple (LCM)
For two non-zero integers:
lcm(a, b) = |a × b| / gcd(a, b)
Use the GCD to compute LCM efficiently
More than Two Numbers
For multiple values, compute pairwise:
gcd(a, b, c) = gcd(gcd(a, b), c)
lcm(a, b, c) = lcm(lcm(a, b), c)
Example Calculations
Example 1: Find GCD of 48 and 18
48 factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
18 factors: 1, 2, 3, 6, 9, 18
Largest common factor: 6
Result: gcd(48, 18) = 6
Example 2: Find LCM of 12 and 18
First find GCD: gcd(12, 18) = 6
Calculation: lcm(12, 18) = |12 × 18| / 6 = 216 / 6 = 36
Result: lcm(12, 18) = 36
Example 3: Simplify a Fraction Using GCD
Problem: Simplify 84/126
GCD: gcd(84, 126) = 42
Calculation: 84 ÷ 42 = 2, 126 ÷ 42 = 3
Result: 84/126 simplifies to 2/3
Example 4: LCM for Multiple Numbers
Problem: Find LCM of 4, 6, 10
Step 1: lcm(4, 6) = 12
Step 2: lcm(12, 10) = 60
Result: lcm(4, 6, 10) = 60
Frequently Asked Questions
What’s the difference between GCD and LCM?
GCD is the largest number that divides both numbers. LCM is the smallest number that both numbers divide into (the smallest shared multiple). GCD helps simplify; LCM helps combine or align values.
When do I use GCD?
Use GCD when simplifying fractions, reducing ratios, or finding the largest equal group size (for example, splitting items evenly).
When do I use LCM?
Use LCM when finding a common denominator for fractions, aligning repeating schedules (like events every 6 and 8 days), or solving problems involving cycles.
Can LCM & GCD be calculated for more than two numbers?
Yes. The standard approach is to compute the result pairwise (combine two numbers at a time) until all numbers are included.
What happens if one of the numbers is 0?
Many definitions treat gcd(a, 0) = |a| and lcm(a, 0) = 0, but calculators may vary. If your tool allows 0, it should clearly define how it handles it.
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What Are LCM and GCD?
The Greatest Common Divisor (GCD) — also called Highest Common Factor (HCF) — is the largest number that divides two or more numbers without a remainder. The Least Common Multiple (LCM) is the smallest positive number that is a multiple of all given numbers. Both are foundational concepts in number theory and play a central role in fraction arithmetic, making them tools you will use throughout math and beyond.
GCD is used to simplify fractions: divide both the numerator and denominator by the GCD to reach the lowest terms. LCM is used to find a common denominator when adding or subtracting fractions with different denominators. This calculator finds both values instantly using the Euclidean algorithm, one of the oldest and most efficient algorithms in mathematics.
How to Use the LCM & GCD Calculator
- Enter two or more integers in the input fields.
- Click Calculate.
- Read the GCD and LCM results displayed below.
- Use the GCD to simplify fractions, or use the LCM to find a common denominator.
Formulas & Algorithms
Euclidean Algorithm (GCD):
GCD(a, b) = GCD(b, a mod b) until b = 0
Example: GCD(48, 18) → GCD(18, 12) → GCD(12, 6) → GCD(6, 0) = 6
LCM from GCD:
LCM(a, b) = |a × b| / GCD(a, b)
Example: LCM(4, 6) = 24 / 2 = 12
Prime factorization method:
48 = 2⁴ × 3, 18 = 2 × 3²
GCD = 2¹ × 3¹ = 6, LCM = 2⁴ × 3² = 144LCM(a, b) × GCD(a, b) = a × b. This identity provides a fast shortcut once you know either value.
Worked Examples
Example 1: GCD(12, 8) and LCM(12, 8)
GCD(12, 8): 12 mod 8 = 4, then GCD(8, 4) = 4. So GCD = 4. LCM = (12 × 8) / 4 = 96 / 4 = 24.
Example 2: Simplify the fraction 18/24
Find GCD(18, 24): 24 mod 18 = 6, then GCD(18, 6) = 6. Divide both by 6: 18/24 = 3/4.
Example 3: Add 1/4 + 1/6
Find LCM(4, 6) = 12. Rewrite: 1/4 = 3/12 and 1/6 = 2/12. Add: 3/12 + 2/12 = 5/12.