What Is Divisibility on the GMAT?

Definition

An integer a is divisible by an integer b (where b ≠ 0) if there exists an integer k such that a = b × k. In other words, dividing a by b yields an integer with no remainder. For example, 12 is divisible by 4 because 12 = 4 × 3, but 12 is not divisible by 5 because 12 ÷ 5 = 2 with remainder 2.

Why It Matters on the GMAT

Divisibility is a core concept in the Number Properties topic. GMAT problems often test divisibility indirectly through:

• Prime factorization and greatest common factor (GCF) / least common multiple (LCM).
• Remainder problems (e.g., “What is the remainder when n is divided by 7?”).
• Determining whether a number is prime, composite, odd, or even.
• Data sufficiency questions that hinge on whether a variable is divisible by a certain number.

Knowing divisibility rules (e.g., a number is divisible by 3 if the sum of its digits is divisible by 3) can save you time on test day.

A Concrete Example

Question: Is the integer n divisible by 6?

Solution: For n to be divisible by 6, it must be divisible by both 2 and 3 (since 6 = 2 × 3). Check the last digit for divisibility by 2 (must be even) and sum the digits for divisibility by 3. For instance, n = 132: last digit is 2 (even), and sum of digits = 1+3+2 = 6, which is divisible by 3. So 132 is divisible by 6 (132 ÷ 6 = 22). If either test fails, n is not divisible by 6.

Common Mistakes

• Confusing divisibility with factors: Saying “a is divisible by b” is the same as “b is a factor of a” – they are equivalent, but phrasing can trip you up.
• Forgetting the zero remainder: Divisibility means no remainder. For example, 7 is not divisible by 2 because 7 ÷ 2 = 3.5, not an integer.
• Applying rules incorrectly: The divisibility rule for 4 is not “last two digits divisible by 4” – that’s correct, but some people mistakenly use the last digit. For 8, it’s the last three digits.
• Assuming divisibility by 6 means divisible by 2 and 3 – that’s true, but for composite numbers, you must check all prime factors.

How to Practice It

To master divisibility, start by memorizing the basic rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11. Then practice with these steps:

• Break down numbers into prime factors – this is the most reliable method.
• Work through GMAT-style problems from official guides or reputable online question banks, focusing on number properties.
• Time yourself: divisibility checks should take seconds, not minutes.
• For data sufficiency, practice rephrasing the question. For example, “Is n divisible by 12?” becomes “Is n divisible by 3 and 4?” (or 2² and 3).

Consistent practice will make divisibility second nature, freeing up mental energy for harder questions.