GCD & LCM

The greatest common divisor is the largest number that divides both exactly, while the least common multiple is the smallest number they both divide into. List the multiples of 12 and 18 and the overlaps are their common multiples; overlay the prime factorizations in a Venn diagram and the shared part is the GCD. The product of the two numbers always equals GCD times LCM, so knowing one lets you find the other at once. Change the two numbers and see where the multiples overlap and which prime factors they share.

Common Divisors & Multiples

A12

B18

👀 Where Multiples Overlap

①List multiples of two numbers — overlaps appear

②Those overlaps are common multiples

③The smallest is the Least Common Multiple (LCM)

Venn Diagram View

🔍 Secret of the Venn Diagram

①Intersection → GCD (Greatest Common Divisor)

②Union → LCM

③GCD = product of common primes raised to the minimum exponent

④LCM = product of all primes raised to the maximum exponent

GCD × LCM Relation

GCD · LCM Relation

a × b = GCD(a,b) × LCM(a,b)

Product of two numbers = GCD × LCM

Computing LCM

LCM(a,b) = a × bGCD(a,b)

Find GCD first, then LCM follows immediately

💡 Computation Order

①Factor both numbers into primes

②Common primes with minimum exponent → GCD

③All primes with maximum exponent → LCM

④Verify: GCD × LCM = a × b

Work It Out

Example 1

Find the greatest common divisor of 12 and 18.

Factor both numbers into primes.

12 = 2² × 3, 18 = 2 × 3²

Multiply the common primes with the lower exponent.

GCD = 2 × 3 = 6

▸ 6

The GCD multiplies common primes at the lower exponent.

Example 2

Find the least common multiple of 12 and 18.

Factor both numbers into primes.

12 = 2² × 3, 18 = 2 × 3²

Multiply every prime with the higher exponent.

LCM = 2² × 3² = 36

▸ 36

The LCM multiplies all primes at the higher exponent.

Exam Key Points

GCD

GCD = product of common primes (min exponents)

Largest number that divides both

LCM

LCM = product of all primes (max exponents)

Smallest common multiple of both

Grade-7 school exam type

What is the side of the largest square tile that exactly covers a 12 cm by 18 cm rectangle?

①2

②3

③6

④9

⑤12

▸ ③ 6

The side of the largest tile that fits exactly is the GCD of the two sides.

side = GCD(12, 18)

Compute the GCD.

= 2 × 3 = 6 (cm)

🎯 Exam Key Points

①GCD use: dividing equally, largest square tile

②LCM use: simultaneous start, smallest common period

③Same idea works for three numbers

④Coprime: GCD = 1

⑤Always verify with GCD × LCM = a × b

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Prime Factorization

Integers & Rationals

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