Question

Is \(\mathrm{n}\) divisible by 3 with no remainder? (1) \(\mathrm{n}-3\) is divisible by 3 with integer result (2) \(\mathrm{n}-21\) is divisible by 3 with integer result A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

D. 1 alone or 2 alone

Step-by-step solution

Analyze Statement 1

Check if - 3 is divisible by 3 with an integer result. If - 3 = 3k for some integer k, then = 3(k + 1) and is divisible by 3.

Conclusion from Statement 1

Since - 3 is divisible by 3, is also divisible by 3. Thus, Statement 1 alone is sufficient.

Analyze Statement 2

Now, check if - 21 is divisible by 3 with an integer result. If - 21 = 3m for some integer m, then = 3(m + 7) and is divisible by 3.

Conclusion from Statement 2

Since - 21 is divisible by 3, is also divisible by 3. Thus, Statement 2 alone is sufficient.

Evaluate Combined Information

Both statements independently prove is divisible by 3. Hence, they do not need to be combined.

Choose the Correct Option

Since each statement independently confirms that is divisible by 3, the answer is D. Either statement alone is sufficient.

Key concepts

GMAT preparation

Preparing for the GMAT requires a good understanding of mathematical concepts and the ability to apply them effectively. To ace the quantitative section, you need to master various problem-solving techniques.

For questions on divisibility, like the one given, it’s crucial to break down each statement and use logical reasoning to determine sufficiency.
Make sure to practice regularly to build confidence and speed.
Familiarize yourself with common divisibility rules and become adept at recognizing patterns to quickly identify whether a number is divisible by another.

number theory

Number theory is a branch of mathematics dealing with the properties and relationships of numbers, especially integers. In this example, understanding how to check divisibility is key.

Here are some essential number theory concepts:

• A number is divisible by 3 if the sum of its digits is divisible by 3.
• If a number minus another specific number is divisible by 3, then the original number is also divisible by 3.

In the given problem, both statements provide different subtractions that still result in multiples of 3, proving that n is divisible by 3. Practicing number theory will enhance your problem-solving skills for the GMAT.

problem-solving skills

Problem-solving skills are essential for tackling GMAT questions effectively. Being systematic in your approach helps in arriving at the correct answer.

Follow these steps:

• Understand the question: What are you trying to prove?
• Break down each statement to see what information it provides independently.

In this example, you need to check the divisibility of n by 3. You analyze each statement separately to see if it alone can confirm the divisibility.

This structure ensures that you comprehensively address each part of the problem without missing crucial details.

quantitative reasoning

Quantitative reasoning involves the ability to work with numerical and logical concepts to solve problems efficiently. This includes recognizing patterns, making generalizations based on given data, and applying mathematical concepts.

In the question about divisibility, you use quantitative reasoning to analyze how subtracting specific numbers (3 and 21) from n affects its divisibility by 3.
If n − 3 and n − 21 are divisible by 3, you logically conclude that n itself must also be divisible by 3. This approach not only saves time but also ensures accuracy. Sharpening your quantitative reasoning skills through practice problems will greatly benefit your GMAT preparation.