Divisibility, Multiples & Factors

Understanding divisibility, multiples, and factors is essential for solving many problems on the GMAT. These concepts are foundational in number theory and are frequently tested in the Quantitative section.

Divisibility

A number $a$ is said to be divisible by another number $b$ if there exists an integer $k$ such that:

$a = b \times k$

In other words, $b$ divides $a$ without leaving a remainder. This is denoted as $b \mid a$.

Multiples

A multiple of a number $b$ is any integer that can be expressed as:

$b \times k$

where $k$ is an integer. For example, the multiples of 3 are 3, 6, 9, 12, etc.

Factors

A factor of a number $a$ is an integer $b$ that divides $a$ without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Prime Factorization

Every integer greater than 1 can be expressed as a product of prime numbers. This is called prime factorization. For example:

$12 = 2^2 \times 3^1$

Prime factorization is useful for finding the greatest common divisor (GCD) and least common multiple (LCM) of two numbers.

Greatest Common Divisor (GCD)

The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Least Common Multiple (LCM)

The LCM of two numbers is the smallest number that is a multiple of both. For example, the LCM of 4 and 6 is 12.

Divisibility Rules

Here are some common divisibility rules:

• Divisible by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisible by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisible by 10: A number is divisible by 10 if its last digit is 0.

Applications in GMAT Problems

These concepts are often tested in problems involving:

• Finding the GCD or LCM of numbers.
• Determining whether a number is divisible by another.
• Solving word problems involving multiples and factors.

Mastering these concepts will help you tackle a wide range of GMAT problems efficiently.