What are the LCM and GCD?
The greatest common divisor is the largest number that divides both values exactly — for 12 and 18 it is 6. The least common multiple is the smallest number both divide into — for 12 and 18 it is 36. They are complementary: GCD strips a pair down to what they share, LCM builds up to the smallest thing that contains both. The diagram shows this as overlapping prime factors.
How do I find them by hand?
For the GCD, the Euclidean algorithm is fastest: divide the larger by the smaller, replace the pair with the smaller and the remainder, repeat until the remainder is zero. For 48 and 18: 48÷18 leaves 12, then 18÷12 leaves 6, then 12÷6 leaves 0, so the GCD is 6. For the LCM, multiply the two numbers and divide by their GCD — 48×18÷6 = 144.
How does prime factorisation give both answers?
Break each number into primes. 12 is 2×2×3 and 18 is 2×3×3. The GCD is the product of the factors they share, counting duplicates — one 2 and one 3, giving 6. The LCM is the product of every factor at its highest power in either number — 2×2×3×3, giving 36. Counting duplicates correctly is the part people get wrong: 8 and 12 share 2×2, not just a single 2.
Is LCM times GCD always equal to the product of the numbers?
For two numbers, yes: GCD × LCM = a × b, always. It is a useful check on your working. Importantly, it does not extend to three or more numbers — for 4, 6 and 10 the GCD is 2 and the LCM is 60, and 2×60 is nowhere near 240. Applying the two-number shortcut to a longer list is a common source of wrong answers.
What are they actually used for?
GCD simplifies fractions: divide numerator and denominator by it and you have the fraction in lowest terms. LCM finds common denominators for adding fractions, and answers scheduling questions — two buses leaving every 12 and 18 minutes coincide every 36 minutes. Beyond arithmetic, GCD underpins modular arithmetic and the Euclidean algorithm is a building block of RSA cryptography.
What if the numbers share no factors?
Then their GCD is 1 and they are called coprime or relatively prime. Their LCM is simply their product — 7 and 13 give a GCD of 1 and an LCM of 91. Coprimality does not require either number to be prime: 8 and 9 are coprime despite both being composite. In the Venn diagram the circles have no overlap at all.