Factor and Multiple GMAT: Key Concepts for Preparation

Key Takeaways

• Definition of Factors and Multiples: Learn the difference between factors, which divide numbers exactly, and multiples, which result from multiplying a number by an integer.
• Prime Factorization: Understand how to break down numbers into prime factors, a method that simplifies complex GMAT questions.
• Greatest Common Factor (GCF) & Least Common Multiple (LCM): Key techniques to solve problems involving GCF and LCM quickly and accurately.
• Common Mistakes: Avoid typical errors students make when working with factors and multiples, such as misinterpreting divisibility rules.
• GMAT-Style Questions: Practice with GMAT-like problems to see how factors and multiples are tested in real exam scenarios.

Understanding factors and multiples is crucial for the GMAT Quantitative section, as these concepts often appear in various question types, from number properties to data sufficiency. If you're aiming for a top score, mastering the fundamentals of factors and multiples will give you the edge needed to solve questions more quickly and efficiently. Whether it's finding the greatest common factor (GCF) or least common multiple (LCM), this guide will provide the insights and tips to help you excel on this critical topic. Let’s dive into the key concepts that will boost your GMAT performance.

What Are Factors and Multiples?

In mathematics, a factor of a number is any integer that divides the number without leaving a remainder. For example, if you consider the number 12, the factors of 12 are: 1, 2, 3, 4, 6, and 12. This means that 12 can be divided exactly by these numbers without producing any fractional results.

One of the most effective ways to find factors is through prime factorization. Prime factorization involves breaking a number down into its prime factors. A prime number is a number that is only divisible by 1 and itself. For instance, the prime factorization of 12 is:

12 = 22 × 3

This breakdown shows that 12 is composed of two 2’s and one 3 as its prime factors. Once you have the prime factorization of a number, you can use it to identify all its factors by considering different combinations of its prime factors.

A multiple is a number that you get by multiplying a given number by an integer. For example, multiples of 5 are: 5, 10, 15, 20, 25, and so on. In simple terms, a multiple of a number is the result of multiplying that number by any whole number.

For instance:

• The multiples of 3 are: 3, 6, 9, 12, 15, etc.
• The multiples of 7 are: 7, 14, 21, 28, 35, etc.

Multiples are endless, meaning there’s no limit to how many multiples of a number can exist, as long as you keep multiplying it by bigger integers. Multiples often appear in GMAT questions where you are asked to find the least common multiple (LCM) of two numbers or check whether a number is divisible by another.

You can refer to Multiple for a detailed analysis of Factor and Multiple GMAT

Key Concepts Related to Factor and Multiple GMAT

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is the largest number that can divide two or more numbers evenly. It’s a crucial concept in the GMAT, especially for questions involving number properties and divisibility.

To find the GCD of two numbers, you use prime factorization. Let’s break it down with an example:

• Find the prime factorization of each number.
• Identify the common prime factors.
• Multiply those common prime factors at their lowest powers.

For instance, let’s calculate the GCD of 36 and 48:

• Prime factorization of 36: 22 × 32
• Prime factorization of 48: 24 × 3

The common prime factors are 2 and 3. We take the lowest powers of these, which are 22 and 31. Thus, the GCD is:

GCD(36, 48) = 22 × 3 = 12

This method is particularly effective when solving GMAT data sufficiency questions that require you to compare multiple numbers.

Another important aspect of GCD is its relationship with the Least Common Multiple (LCM). There’s a direct formula to relate GCD and LCM for any two numbers a and b:

GCD(a, b) × LCM(a, b) = a × b

This relationship is tested in various Factor and Multiple GMAT questions, so understanding it can save you valuable time.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. Just like the GCD, the LCM is essential for solving number property questions on the GMAT. The LCM is particularly useful when you're dealing with problems that ask how often events will coincide, or how to manage numbers that have different divisibility rules.

To find the LCM, follow these steps:

1. Break down the numbers into their prime factors.
2. For each prime factor, take the highest power found in any of the numbers.
3. Multiply all the highest powers together to get the LCM.

For example, let’s calculate the LCM of 36 and 48:

• Prime factorization of 36: 22 × 32
• Prime factorization of 48: 24 × 3

We take the highest powers of each prime factor: 24 and 32. So, the LCM is:

LCM(36, 48) = 24 × 32 = 144

Knowing the LCM helps solve problems that ask when two or more tasks will coincide, or when two numbers will divide evenly into another number.

Here’s a summary of GCD vs. LCM:

You can refer GMAT Math: Factors - Prime Numbers for detailed analysis of Factor and Multiple GMAT

How Factors and Multiples Appear on the GMAT

Data Sufficiency Questions

Factors and multiples often come up in data sufficiency questions on the GMAT. In these questions, you are asked whether a statement provides enough information to answer a question. Let’s look at an example:

Is m a multiple of 15?

• (1) m is divisible by 5.
• (2) m is divisible by 3.

By the rule of factors and multiples, if a number is divisible by both 5 and 3, then it must also be divisible by their product, which is 15. In this case, the statements together are sufficient to determine that m is a multiple of 15.

Other common types of data sufficiency questions involve finding the greatest common divisor or least common multiple of two numbers. In these cases, you often need to decide whether the information given is sufficient to make these calculations.

Problem-Solving Questions

Problem-solving questions frequently involve factors and multiples, especially in the number properties section of the GMAT. These questions ask you to either calculate the GCD, the LCM, or determine if one number is a factor or multiple of another.

For instance:

What is the LCM of 8 and 12?

• Prime factorization of 8: 23
• Prime factorization of 12: 22 × 3

Take the highest powers of each prime factor: 23 and 31. So, the LCM is:

LCM(8, 12) = 23 × 3 = 24

These types of problems are common in the GMAT because they test your ability to apply number theory concepts quickly and accurately.

Another type of problem might ask you to find how many factors a number has. To find the number of factors, follow these steps:

1. Find the prime factorization of the number.
2. Add 1 to each exponent.
3. Multiply the results together.

For example, let’s find how many factors 60 has:

• Prime factorization: 22 × 31 × 51

Add 1 to each exponent: (2 + 1) × (1 + 1) × (1 + 1) = 3 × 2 × 2 = 12

Thus, 60 has 12 factors.

Strategies for Tackling Factor and Multiple Questions on GMAT

Prime Factorization Method

Using the prime factorization method is one of the most effective strategies for solving GMAT problems involving factors and multiples. This method allows you to simplify complex problems and make calculations easier. Here’s how to apply it:

1. Break Down the Numbers: Start by writing the prime factorization of each number involved. This helps you see the basic building blocks of the numbers.
2. Identify Common Factors: For GCD problems, look for the common prime factors and choose their lowest powers.
3. Identify Highest Powers: For LCM problems, take the highest powers of all prime factors present in the numbers.
4. Calculate: Multiply the identified factors together to get the GCD or LCM as required.

Example: To find the GCD and LCM of 18 and 30:

• Prime factorization of 18: 21 × 32
• Prime factorization of 30: 21 × 31 × 51

GCD: Take the lowest powers of common factors: 21 × 31 = 6

LCM: Take the highest powers of all factors: 21 × 32 × 51 = 90

This method streamlines the process and helps avoid errors in calculations.

Divisibility Rules

Knowing the divisibility rules is crucial for quickly solving GMAT questions involving factors and multiples. These rules help determine if one number is a factor of another without performing long division. Here are some essential rules:

• Divisible by 2: If the last digit is even (0, 2, 4, 6, 8).
• Divisible by 3: If the sum of the digits is divisible by 3.
• Divisible by 4: If the last two digits form a number that is divisible by 4.
• Divisible by 5: If the last digit is 0 or 5.
• Divisible by 6: If the number is divisible by both 2 and 3.
• Divisible by 10: If the last digit is 0.

These rules can save time on the GMAT, especially in data sufficiency and problem-solving questions.

Practice Questions and Solutions

Sample GMAT Questions on Factors

1. Question: What is the GCD of 48 and 64?
   • Prime factorization of 48: 24 × 31
   • Prime factorization of 64: 26
   • Answer: GCD is 24 = 16
2. Question: How many factors does 36 have?
   • Prime factorization: 22 × 32
   • Calculation: (2 + 1)(2 + 1) = 3 × 3 = 9
   • Answer: 36 has 9 factors.

Sample GMAT Questions on Multiples

1. Question: What is the LCM of 15 and 20?
   • Prime factorization of 15: 31 × 51
   • Prime factorization of 20: 22 × 51
   • Calculation: LCM is 22 × 31 × 51 = 60
   • Answer: LCM is 60.
2. Question: Is 42 a multiple of 7?
   • Calculation: 42 ÷ 7 = 6, which is an integer.
   • Answer: Yes, 42 is a multiple of 7.

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Conclusion 

Understanding factors and multiples is essential for success on the GMAT, particularly in the quantitative section. Mastering these concepts not only helps in solving specific questions but also enhances your overall problem-solving skills. Factors play a critical role in determining divisibility, while multiples are crucial for solving problems related to common occurrences and least common multiples.