2.2 Greatest Common Factor (GCF) & Least Common Multiple (LCM)

The greatest common factor (GCF), also called the greatest common divisor (GCD), of two or more integers is the largest positive integer that divides every one of them. The least common multiple (LCM) is the smallest positive integer that is a multiple of every one of them.

On GMAT Problem Solving, GCF is the tool for “largest equal groups with nothing left over” and “largest square tile that fills a rectangle.” LCM is the tool for “when do repeating cycles next coincide” and “smallest number that is a multiple of each of these integers.” Confusing GCF with LCM is one of the highest-yield traps in Math Review 3.1.

Prime-factor method

Write each integer in prime factorization, using exponent 0 for a prime that is missing.

For a = p1^u1 × p2^u2 × … and b = p1^v1 × p2^v2 × …:

• GCF(a, b) uses the minimum exponent of each prime.
• LCM(a, b) uses the maximum exponent of each prime.

A Venn diagram of the prime-factor multisets makes the same rule visual. Shared primes with the smaller exponent sit in the overlap (GCF). The union of all prime powers, using the larger exponent of each prime, is the LCM.

Worked example: 360 and 840

360 = 36 × 10 = 2^3 × 3^2 × 5

840 = 84 × 10 = 2^3 × 3 × 5 × 7

GCF(360, 840) = 2^3 × 3 × 5 = 8 × 3 × 5 = 120

LCM(360, 840) = 2^3 × 3^2 × 5 × 7 = 8 × 9 × 5 × 7 = 2,520

Check with the product identity: 360 × 840 = 302,400 and 120 × 2,520 = 302,400.

A smaller Venn you should be able to draw in 20 seconds: 12 = 2^2 × 3 and 18 = 2 × 3^2. The overlap is one 2 and one 3, so GCF = 6. The union is 2^2 × 3^2, so LCM = 36. And 6 × 36 = 216 = 12 × 18.

The product identity

For any two positive integers a and b:

GCF(a, b) × LCM(a, b) = a × b

For any two non-zero integers the same relation is GCF × LCM = |ab|. GMAT Quant almost always gives positive integers, so you can drop the absolute values.

Why it works: for each prime, min(u, v) + max(u, v) = u + v, so the exponents on the left reconstruct the product a × b.

Trap — three or more integers: GCF(a, b, c) × LCM(a, b, c) is not equal to a × b × c. Counterexample: GCF(2, 4, 8) = 2 and LCM(2, 4, 8) = 8, so the product of GCF and LCM is 16, but 2 × 4 × 8 = 64.

If GCF(a, b) = 1, the integers are relatively prime (coprime), and LCM(a, b) = a × b. Consecutive integers are always coprime, so LCM(n, n+1) = n(n+1).

Euclidean algorithm (no calculator)

When factorization is ugly, use remainders:

GCF(a, b) = GCF(b, remainder when a is divided by b), and GCF(a, 0) = |a|.

Example: GCF(744, 216)

• 744 = 216 × 3 + 96, so GCF(744, 216) = GCF(216, 96)
• 216 = 96 × 2 + 24, so GCF(216, 96) = GCF(96, 24)
• 96 = 24 × 4 + 0, so GCF = 24

Then LCM(744, 216) = (744 × 216) / 24 if you need the companion multiple. On paper, cancel 24 into 744 first: 31 × 216 = 6,696.

Algebraic monomials

The same min/max rules apply to variable powers.

• GCF(12 x^3 y^2 z, 18 x^2 y^4) = 6 x^2 y^2
• LCM(12 x^3 y^2 z, 18 x^2 y^4) = 36 x^3 y^4 z

Coefficients use ordinary numeric GCF/LCM; each variable takes min or max exponent independently. A variable present in only one monomial has min exponent 0 (it drops out of the GCF) and max equal to the exponent that is present (it appears in the LCM).

Word-problem archetypes

Repeating events: use LCM

Three lights flash every 12, 18, and 30 seconds and flash together at 12:00. Next simultaneous flash:

12 = 2^2 × 3, 18 = 2 × 3^2, 30 = 2 × 3 × 5

LCM = 2^2 × 3^2 × 5 = 180 seconds = 3 minutes, so they next flash together at 12:03.

Two runners who start together at a starting line and complete laps in 72 seconds and 108 seconds next occupy that starting line at the same moment at LCM(72, 108). That is not GCF(72, 108) = 36, which is when the faster runner has gained full extra laps in a relative-speed story — a different question. Read whether the stem asks for a common arrival at a marked point (LCM of the periods) or for a largest equal grouping (GCF).

Tiling and equal groups: use GCF

A 72 cm by 120 cm board is cut into identical squares with no waste. Largest square side = GCF(72, 120) = 24 cm. Number of squares = (72/24) × (120/24) = 3 × 5 = 15. If the question asks for the minimum number of uncut identical squares, you still use the largest side, because larger tiles mean fewer tiles.

Consecutive common multiples

The common multiples of 12 and 18 are exactly the multiples of LCM(12, 18) = 36. The positive common multiples ≤ 200 are 36, 72, 108, 144, 180 — that is floor(200/36) = 5 values.

In general:

• The number of positive multiples of k that are ≤ N is floor(N/k).
• The number of multiples of k from A through B inclusive is floor(B/k) − floor((A−1)/k).

Trap: “how many multiples of 6 from 1 to 100” is floor(100/6) = 16, not 100/6 rounded up, and not 17. “How many common multiples of 4 and 6 from 1 to 100” is the number of multiples of LCM(4, 6) = 12, which is floor(100/12) = 8.

GCF of more than two numbers

GCF(a, b, c) = GCF(GCF(a, b), c). You still take the minimum exponent across all of the integers. LCM(a, b, c) takes the maximum across all.

Example: GCF(24, 36, 60). Factor 24 = 2^3 × 3, 36 = 2^2 × 3^2, 60 = 2^2 × 3 × 5. Minimum exponents give 2^2 × 3 = 12.

Choosing GCF vs LCM under GMAT wording

If a question gives GCF and LCM of two positives and one of the integers, the other integer is (GCF × LCM) / known integer. Always check that the stated GCF actually divides both answers; a choice that is not a multiple of the given GCF cannot be correct.