Solving GRE Number Properties Without Using Big Calculations

Imagine you are faced with a GRE question asking whether x(y + 1) is even or odd. Many students immediately start plugging in random numbers like 2 and 3, which works for one case but often leads to a trap if 0 or negative integers are ignored. Number properties are not about performing long division or heavy multiplication. Instead, they are about the structural DNA of integers. A single even factor in a long string of multiplication guarantees an even result, regardless of how large the other numbers are. Recognizing these inherent behaviors allows you to bypass the calculator entirely.

The difficulty in these easy-level questions usually lies in the wording, particularly the distinction between must be and could be. When the GRE asks what must be true, you are looking for a property that holds up under every possible integer constraint, including prime numbers and consecutive sequences. For instance, in any set of three consecutive integers, exactly one of those numbers will be divisible by three. This article focuses on identifying these mechanical rules so you can spot the correct answer choice in seconds rather than minutes.

The Logic of Integer Behavior

Number properties refer to the set of rules that describe the behavior of integers, including concepts like parity, divisibility, and prime factorization. On the GRE, these questions test your ability to recognize patterns rather than perform heavy computation. Key concepts include:

• Parity: The classification of an integer as either even or odd. Remember that $\text{Even} \times \text{Any integer} = \text{Even}$, and $\text{Odd} \times \text{Odd} = \text{Odd}$.
• Divisibility: An integer $n$ is divisible by $d$ if the remainder is zero. Common shortcuts include checking if the sum of digits is divisible by 3 or 9.
• Prime Numbers: Integers greater than 1 that have exactly two factors: 1 and themselves. Note that 2 is the only even prime number.
• Remainders: The amount left over after division. If $x$ is divided by $y$, the remainder $r$ must satisfy $0 \leq r < y$.
• Consecutive Integers: A sequence of integers following one another, such as $n, n+1, n+2$. In any set of $k$ consecutive integers, exactly one is divisible by $k$.

For more practice with various question types, you can explore Free GRE Practice Questions Practice Questions with Answers to sharpen your skills.

Solved Examples

1. If $x$ is an even integer and $y$ is an odd integer, which of the following must be an even integer?

1. Identify the properties: $x = 2n$ (even) and $y = 2m + 1$ (odd).
2. Test the operations: $x + y = \text{Even} + \text{Odd} = \text{Odd}$.
3. Test multiplication: $x \times y = \text{Even} \times \text{Odd} = \text{Even}$.
4. Conclusion: Since any integer multiplied by an even number results in an even product, $xy$ is the correct choice.

2. What is the greatest common factor (GCF) of 24 and 60?

1. Find prime factors of 24: $24 = 2^3 \times 3$.
2. Find prime factors of 60: $60 = 2^2 \times 3 \times 5$.
3. Identify common factors: Both share $2^2$ and $3^1$.
4. Multiply common factors: $4 \times 3 = 12$. The GCF is 12.

3. If $k$ is a prime number and $17 < k < 25$, what is the value of $k$?

1. List integers between 17 and 25: 18, 19, 20, 21, 22, 23, 24.
2. Eliminate even numbers: 19, 21, 23.
3. Eliminate multiples of 3: 21 is $3 \times 7$, so we are left with 19 and 23.
4. Check for primality: Both 19 and 23 have no other factors. In a GRE multiple-choice context, you would select the one provided in the options.

Practice Questions

1. If $n$ is an integer, which of the following must be odd?

2. What is the remainder when 45 is divided by 7?

3. Which of the following is a prime number: 27, 37, 47, or 57?

4. If $x$ is a multiple of 6 and $y$ is a multiple of 9, then $x + y$ must be a multiple of which integer?

5. How many prime numbers are there between 10 and 20?

6. If $a$ and $b$ are positive integers such that $a - b$ is even, which of the following must be true?

7. What is the least common multiple (LCM) of 8 and 12?

8. If $m$ is an even integer, is $m^2 + m + 1$ even or odd?

9. A number $z$ is divisible by both 4 and 15. What is the smallest possible positive value for $z$?

10. If the sum of three consecutive integers is 45, what is the largest of these integers?

Answers & Explanations

1. Answer: $2n + 1$. Any integer multiplied by 2 becomes even; adding 1 to an even number always results in an odd number.

2. Answer: 3. Since $7 \times 6 = 42$, the remainder is $45 - 42 = 3$. You can use the AI Question Generator to create more remainder problems for practice.

3. Answer: 37 and 47. 27 is $3 \times 9$; 57 is $3 \times 19$. Both 37 and 47 are prime. According to the Wikipedia definition of primes, they have no divisors other than 1 and themselves.

4. Answer: 3. $x = 6k$ and $y = 9j$. Their sum is $3(2k + 3j)$, which is always divisible by 3. For similar logic puzzles, check GRE Text Completion Exam Questions Practice Questions with Answers.

5. Answer: 4. The prime numbers between 10 and 20 are 11, 13, 17, and 19.

6. Answer: $a$ and $b$ have the same parity. For the difference of two integers to be even, they must both be even or both be odd.

7. Answer: 24. Multiples of 8: 8, 16, 24, 32. Multiples of 12: 12, 24. The smallest common multiple is 24.

8. Answer: Odd. If $m$ is even, $m^2$ is even. $\text{Even} + \text{Even} = \text{Even}$. Then, $\text{Even} + 1 = \text{Odd}$.

9. Answer: 60. This is the LCM of 4 and 15. Since they share no common factors, $4 \times 15 = 60$.

10. Answer: 16. Let the integers be $n-1, n, n+1$. Their sum is $3n = 45$, so $n = 15$. The largest is $15 + 1 = 16$. For more strategy on word problems, visit GRE Writing Strategy Questions Practice Questions with Answers.

Frequently Asked Questions

Is 1 a prime number on the GRE?

No, 1 is not considered a prime number because a prime number must have exactly two distinct factors: 1 and itself. Since 1 only has one factor, it is excluded from the list of primes.

How do I quickly tell if a large number is divisible by 3?

Add all the individual digits of the number together. If the resulting sum is divisible by 3, then the original large number is also divisible by 3.

What is the difference between a factor and a multiple?

A factor is a number that divides into another number evenly, while a multiple is the product of a number and an integer. For example, 3 is a factor of 12, and 12 is a multiple of 3.

Are negative numbers considered even or odd?

Yes, parity applies to all integers, including negative ones. For example, -2 and -4 are even, while -1 and -3 are odd, following the same alternating pattern as positive integers.

What is the rule for the sum of two odd numbers?

The sum of two odd numbers is always even. This can be represented algebraically as $(2n+1) + (2m+1) = 2n + 2m + 2$, which is clearly divisible by 2.

Can a remainder be larger than the divisor?

No, a remainder must always be a non-negative integer that is strictly less than the divisor. If you calculate a remainder equal to or larger than the divisor, the division process is not yet complete.