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Greatest common divisor and least common multiple
Find the largest integer dividing both numbers and the smallest positive multiple shared by them. These answer different grouping and cycle questions.
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Greatest common divisor
6
A common divisor divides both numbers; a common multiple is divisible by both numbers.
Least common multiple36
Integer ratio in lowest terms2:3
Worked steps
- 12 ÷ 18 = 0, remainder 12
- 18 ÷ 12 = 1, remainder 6
- 12 ÷ 6 = 2, remainder 0
- The last nonzero divisor is 6; least common multiple = 12 ÷ 6 × 18 = 36
Enter two positive integers from 1 to 1000000. Uses Euclid's algorithm for GCD; LCM = first number ÷ GCD × second number. Zero, negatives, decimals and more than two numbers are unsupported. Convert recurring-event periods to the same unit first and assume both begin at the same time.
Method and scope
The greatest common divisor uses the Euclidean algorithm. The least common multiple equals the first number divided by the GCD, times the second number.
GCD can describe the size of equal groups with no remainder. LCM can describe a shared repeat period only when both cycles use the same unit and start together.
Enter two positive integers from 1 to 1,000,000. Zero, negatives, decimals and lists of three or more numbers are outside scope.
References:National Taiwan Normal University number theory notes: GCD and LCM
What this helps you do
Find the largest integer dividing both numbers and the smallest positive multiple shared by them. These answer different grouping and cycle questions.
What to enter
Start with first positive integer, then fill the other relevant fields in the units shown. The field hints explain the required format and examples.
How the calculation works
The greatest common divisor uses the Euclidean algorithm. The least common multiple equals the first number divided by the GCD, times the second number.
Worked example
For 12 and 18, the GCD is 6 and the LCM is 36. Two cycles starting together and repeating every 12 and 18 minutes next coincide after 36 minutes.
Understanding the answer
GCD can describe the size of equal groups with no remainder. LCM can describe a shared repeat period only when both cycles use the same unit and start together.
Scope and limitations
Enter two positive integers from 1 to 1,000,000. Zero, negatives, decimals and lists of three or more numbers are outside scope.