GCD and LCM Calculator

The Greatest Common Divisor (GCD) is the largest number that divides both integers without a remainder. The Least Common Multiple (LCM) is the smallest number that both integers divide into evenly. These are fundamental tools in fraction simplification, scheduling problems, and number theory.

• Simplifying fractions: Divide numerator and denominator by their GCD to reduce to lowest terms. 48/18 → 8/3.
• Finding common denominators: LCM of the denominators is the least common denominator for adding fractions.
• Scheduling problems: Two events repeat every 6 and 8 days. They next coincide after LCM(6,8) = 24 days.
• Tile and grid problems: Find the largest square tile that fits evenly in a rectangular room using GCD of dimensions.

1. 1Enter two positive integers.
2. 2The calculator applies the Euclidean algorithm: divide the larger by the smaller, take the remainder, repeat.
3. 3When the remainder is zero, the last divisor is the GCD.
4. 4LCM is computed from LCM = (a × b) / GCD.

• Expecting GCD to work with decimals or fractions: The GCD is defined for integers only. If you have 2.5 and 5, convert to a common form first (e.g., multiply both by 2 to get 5 and 10, GCD = 5). The calculator requires whole numbers.
• Confusing GCD with LCM: GCD is the largest number that divides both evenly. LCM is the smallest number both divide into. GCD is used to simplify fractions; LCM is used to find common denominators when adding them.
• Entering zero: GCD(0, n) = n by mathematical convention. Some calculators handle this; others do not. If you enter zero, the result is the other number. This is correct but rarely what you intended.
• Entering negative numbers: GCD is defined for positive integers. For negative inputs, most algorithms use the absolute value. Check that your calculator returns the expected result if you enter negative numbers.

• Use GCD to simplify fractions instantly: Divide both numerator and denominator by their GCD to reduce a fraction to lowest terms. GCD(18, 24) = 6, so 18/24 = 3/4.
• Use LCM to find common denominators: To add 1/6 + 1/4, find LCM(6, 4) = 12. Then convert both fractions to twelfths: 2/12 + 3/12 = 5/12.
• Verify by testing divisibility: If the GCD of 48 and 60 is 12, then 48 ÷ 12 = 4 and 60 ÷ 12 = 5 should both be whole numbers. If not, the GCD is wrong.
• Use for scheduling problems: Two events that repeat every 8 and 12 days will next occur on the same day after LCM(8, 12) = 24 days. LCM solves alignment and synchronization problems in scheduling, gears, and timing systems.