GMAT Online Course: Quantitative Skills LCM and GCD

Goalisb
Aug 6, 2025
7 min read

Finding Common Ground: Mastering LCM and GCD for GMAT Success

In our exploration of number properties, few concepts are as practically significant for the GMAT as the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF). These two ideas are direct applications of our understanding of factors and multiples, providing powerful tools to solve a range of quantitative problems, from simplifying fractions to tackling complex word problems involving cycles, common events, or equitable distribution. A firm grasp of how to find and apply LCM and GCD, especially through efficient methods like prime factorization, is indispensable for GMAT mastery.

I. Greatest Common Divisor (GCD) / Highest Common Factor (HCF)

The GCD of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. Think of it as the biggest "shared building block" for those numbers.

Formal Definition: The Greatest Common Divisor (GCD) of two or more integers is the largest positive integer that divides all of them evenly.

Methods to Find the GCD:

Listing Factors Method (Best for Smaller Numbers):
- List all positive factors of each number.
- Identify the common factors.
- The largest among the common factors is the GCD.
Example: Find the GCD of 12 and 18.
- Factors of 12: {1, 2, 3, 4, 6, 12}
- Factors of 18: {1, 2, 3, 6, 9, 18}
- Common Factors: {1, 2, 3, 6}
- The largest common factor is 6. So, GCD(12, 18) = 6.
Prime Factorization Method (Most Efficient for Larger Numbers):
This method is the most powerful and reliable, especially for larger numbers or more than two numbers.
- Find the prime factorization of each number.
- Identify all prime factors that are common to all the numbers.
- For each common prime factor, take the one with the lowest power (exponent).
- Multiply these selected prime factors together.
Example: Find the GCD of 72 and 108.
- Prime factorization of 72: 72 = 2^3 3^2 (i.e., 2 2 2 3 * 3)
- Prime factorization of 108: 108 = 2^2 3^3 (i.e., 2 2 3 3 * 3)
- Common prime factors are 2 and 3.
- Lowest power of 2: 2^2
- Lowest power of 3: 3^2
- GCD = 2^2 3^2 = 4 9 = 36.

Properties of GCD:

GCD(A, B) is always less than or equal to the smaller of A and B.
If A is a factor of B, then GCD(A, B) = A. (e.g., GCD(5, 20) = 5).
Coprime (Relatively Prime) Numbers: Two numbers are coprime if their GCD is 1. (e.g., GCD(7, 15) = 1). This is an important concept in number theory.

II. Least Common Multiple (LCM)

The LCM of two or more integers is the smallest positive integer that is a multiple of all the1 given integers. Think of it as the first point where their "counting patterns" intersect.

Formal Definition:

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of all of them.2

Methods to Find the LCM:

Listing Multiples Method (Best for Smaller Numbers):
- List the first few multiples of each number.
- Identify the common multiples.
- The smallest among the common multiples is the LCM.
Example: Find the LCM of 6 and 8.
- Multiples of 6: {6, 12, 18, 24, 30, 36, ...}
- Multiples of 8: {8, 16, 24, 32, 40, ...}
- The smallest common multiple is 24. So, LCM(6, 8) = 24.3
Prime Factorization Method (Most Efficient for Larger Numbers):
- Find the prime factorization of each number.
- Identify all prime factors present in any of the numbers (common and uncommon).
- For each prime factor, take the one with the highest power (exponent).
- Multiply these selected prime factors together.
Example: Find the LCM of 72 and 108.
- Prime factorization of 72: 72 = 2^3 * 3^2
- Prime factorization of 108: 108 = 2^2 * 3^3
- All prime factors present are 2 and 3.
- Highest power of 2: 2^3
- Highest power of 3: 3^3
- LCM = 2^3 3^3 = 8 27 = 216.

Properties of LCM:

LCM(A, B) is always greater than or equal to the larger of A and B.
If A is a factor of B, then LCM(A, B) = B. (e.g., LCM(5, 20) = 20).

III. The Fundamental Relationship Between LCM and GCD

For any two positive integers A and B, there is a powerful relationship between their LCM and GCD:

LCM(A, B) GCD(A, B) = A B

This formula is incredibly useful on the GMAT, especially if you know two of the values and need to find the third.

Example: Using 12 and 18 again:

GCD(12, 18) = 6
LCM(12, 18) (from listing multiples) = 36
A B = 12 18 = 216
LCM GCD = 36 6 = 216
The formula holds: 216 = 216.

IV. Applications of LCM and GCD (Word Problems on the GMAT)

Recognizing when to use GCD or LCM in word problems is a key GMAT skill.

When to use GCD (Finding the "Largest Group" or "Breaking into Smallest Parts"):

Dividing/Cutting: Problems that ask for the largest possible size of smaller, identical pieces that can be cut from larger pieces without waste.
- Example: You have two ropes, one 24 feet long and the other 36 feet long. What is the greatest length you can cut both ropes into so that you have no leftover rope? (Find GCD(24, 36)). Answer: 12 feet.
Grouping/Arranging: Problems asking for the largest number of identical groups that can be formed from different sets of items.
- Example: A teacher has 48 pencils and 60 erasers. She wants to divide them into equal groups for her students, with no items left over. What is the greatest number of groups she can make? (Find GCD(48, 60)). Answer: 12 groups.

When to use LCM (Finding the "First Time Something Happens Together" or "Common Cycle"):

Cycles/Events: Problems involving events that repeat at different intervals and you need to find when they will occur simultaneously again.
- Example: Two buses leave the station at 8:00 AM. Bus A leaves every 15 minutes, and Bus B leaves every 20 minutes. At what time will they next leave the station at the same time? (Find LCM(15, 20)). Answer: LCM is 60 minutes. So, 9:00 AM.
Finding Smallest Common Quantity: Problems asking for the smallest number that can be exactly divided by several other numbers.
- Example: What is the smallest number of candies that can be evenly divided among 6, 8, or 10 children? (Find LCM(6, 8, 10)). Answer: 120 candies.
Fractions: Finding the Least Common Denominator (LCD) when adding or subtracting fractions is an application of LCM.

V. Common Pitfalls to Avoid

Confusing GCD and LCM: This is the most common error. Remember: GCD is about the largest shared factor (usually smaller than or equal to the numbers), while LCM is about the smallest shared multiple (usually larger than or equal to the numbers).
Prime Factorization Errors: Mistakes in factoring numbers into primes will lead to incorrect LCM/GCD.
Using Incorrect Exponents: For GCD, use the lowest power of common prime factors. For LCM, use the highest power of all prime factors (common and uncommon).
Applying A B = LCM GCD to more than two numbers: This formula is valid only for two numbers.
Not including all primes for LCM: Make sure to account for all unique prime factors from any of the numbers when finding the LCM by prime factorization.

Interactive Check Your Understanding:

Find the GCD of 45 and 75.
Find the LCM of 14 and 21.
Given GCD(A, B) = 5 and A * B = 100, what is LCM(A, B)?
If you want to arrange 30 red marbles and 42 blue marbles into bags, with each bag containing the same number of red marbles and the same number of blue marbles, what is the maximum number of bags you can make? Which concept (LCM or GCD) applies here?

Practice Questions:

What is the least common multiple of 24, 30, and 40?
a) 60
b) 120
c) 240
d) 360
Two wires are 108 meters and 72 meters long. They are to be cut into pieces of equal length, such that each piece is as long as possible. What is the length of each piece?
The product of two positive integers is 360. If their greatest common divisor is 6, what is their least common multiple?
Three friends, Alex, Ben, and Chloe, jog around a circular track. Alex completes a lap in 4 minutes, Ben in 6 minutes, and Chloe in 8 minutes. If they all start at the same point and time, after how many minutes will they all be at the starting point together again for the first time?
Which of the following pairs of numbers has a GCD of 8 and an LCM of 120?
a) 8 and 15
b) 24 and 40
c) 32 and 60
d) 16 and 30

Solutions to Practice Questions:

LCM of 24, 30, and 40:
- Prime factorize:
  - 24 = 2^3 * 3
  - 30 = 2 3 5
  - 40 = 2^3 * 5
- Take highest powers of all prime factors (2, 3, 5):
  - 2^3 (from 24 or 40)
  - 3^1 (from 24 or 30)
  - 5^1 (from 30 or 40)
- LCM = 2^3 3 5 = 8 3 5 = 120. The correct answer is b) 120.
Length of each piece (two wires, 108m and 72m, equal length, as long as possible):
This is a GCD problem.
- Prime factorize:
  - 108 = 2^2 * 3^3
  - 72 = 2^3 * 3^2
- Take lowest powers of common prime factors (2, 3):
  - 2^2 (from 108)
  - 3^2 (from 72)
- GCD = 2^2 3^2 = 4 9 = 36 meters.
Product = 360, GCD = 6. Find LCM:
Use the formula: LCM(A, B) GCD(A, B) = A B
LCM * 6 = 360
LCM = 360 / 6 = 60.
When will they all be at the starting point together again (Alex 4 min, Ben 6 min, Chloe 8 min)?
This is an LCM problem.
- Prime factorize:
  - 4 = 2^2
  - 6 = 2 * 3
  - 8 = 2^3
- Take highest powers of all prime factors (2, 3):
  - 2^3 (from 8)
  - 3^1 (from 6)
- LCM = 2^3 3 = 8 3 = 24 minutes.
Pair with GCD=8 and LCM=120:
Use the formula A B = LCM GCD.
So, A B must equal 120 8 = 960.
Let's check the options:
a) 8 and 15: 8 * 15 = 120 (Incorrect product).
b) 24 and 40: 24 * 40 = 960. This fits the product requirement.
* Now, check GCD(24, 40):
24 = 2^3 3
40 = 2^3 5
* GCD = 2^3 = 8. This matches.
* Check LCM(24, 40):
LCM = 2^3 3 5 = 8 3 * 5 = 120. This matches.
c) 32 and 60: 32 * 60 = 1920 (Incorrect product).
d) 16 and 30: 16 * 30 = 480 (Incorrect product).
The correct answer is b) 24 and 40.