What is the least common multiple (LCM)?+
The least common multiple (LCM) is the smallest positive integer that is divisible by all given numbers. LCM(4, 6) = 12 because 12 is the smallest number that both 4 and 6 divide into evenly. It is also called the Lowest Common Multiple or, when used with fractions, the Least Common Denominator (LCD).
How do you find the LCM of two numbers?+
The fastest method uses GCF: LCM(a, b) = (a × b) / GCF(a, b). Step 1: find GCF using the Euclidean algorithm. Step 2: divide the product by GCF. Example: LCM(12, 18): GCF = 6; LCM = (12 × 18) / 6 = 216 / 6 = 36. Verify: 36 / 12 = 3 and 36 / 18 = 2, both exact integers.
What is LCM(4, 6)?+
LCM(4, 6) = 12. The multiples of 4 are 4, 8, 12, 16, 20... The multiples of 6 are 6, 12, 18, 24... The first common value is 12. Via formula: GCF(4, 6) = 2; LCM = (4 × 6) / 2 = 24 / 2 = 12. LCM(4, 6) is used as the common denominator when adding 1/4 + 1/6.
What is the difference between LCM and GCF?+
GCF (Greatest Common Factor) is the largest number that divides all inputs exactly. LCM is the smallest number that all inputs divide exactly. GCF simplifies (fractions), LCM scales up (common denominators). They are linked: GCF(a,b) × LCM(a,b) = a × b. For a=12, b=18: GCF=6, LCM=36, and 6 × 36 = 216 = 12 × 18.
How do you find LCM using prime factorization?+
Write each number as a product of prime factors. Take each distinct prime base and raise it to its highest exponent across all numbers, then multiply. Example: LCM(12, 18, 20): 12=2²×3; 18=2×3²; 20=2²×5. Primes with highest exponents: 2², 3², 5. LCM = 4 × 9 × 5 = 180. Verify: 180/12=15, 180/18=10, 180/20=9, all exact.
How is LCM used to add fractions?+
To add fractions with different denominators, find the LCM of the denominators (the LCD). Convert each fraction to an equivalent fraction with that denominator, then add the numerators. To add 1/4 + 1/6: LCD = LCM(4,6) = 12. Convert: 1/4 = 3/12 and 1/6 = 2/12. Sum: 5/12. Using LCM rather than a larger common denominator keeps the result in lowest terms automatically.
What is the LCM of two consecutive numbers?+
The LCM of any two consecutive integers n and n+1 equals their product n × (n+1), because consecutive integers are always coprime (GCF = 1). For example: LCM(7, 8) = 56; LCM(15, 16) = 240; LCM(99, 100) = 9,900. This is why fractions with consecutive denominators (like 1/7 + 1/8) produce relatively large common denominators.
How do you find LCM of three or more numbers?+
Apply LCM iteratively. Find LCM of the first two numbers, then find LCM of that result and the third number, continuing until all numbers are included. LCM(4, 6, 10): LCM(4, 6) = 12; LCM(12, 10) = 60. Result: 60. This works because LCM is associative: LCM(a, b, c) = LCM(LCM(a, b), c).
What happens to LCM when one number divides the other?+
If one number is a multiple of the other, their LCM equals the larger number. LCM(6, 12) = 12 because 12 is already a multiple of 6. LCM(5, 25) = 25. LCM(a, b) = b whenever b is divisible by a. This is the simplest case: the larger number is already the common multiple.
What is LCM used for in real life?+
LCM solves synchronization and scheduling problems. When do two events with different cycles next coincide? If a security camera rotates on a 8-second cycle and a motion sensor resets every 12 seconds, LCM(8, 12) = 24 seconds. Bus schedules, gear alignments, planetary conjunctions, and repeating task schedules all use LCM. In construction, LCM finds the shortest repeating tile pattern length when mixing tiles of different widths.
Is the LCM always larger than the original numbers?+
Yes, LCM(a, b) is always greater than or equal to max(a, b). It equals max(a, b) only when one number divides the other (e.g., LCM(4, 12) = 12). Otherwise LCM is strictly larger than both. The LCM can never be smaller than either input because it must be divisible by both, which requires it to be at least as large as each.