Hard GRE Arithmetic Practice Test Practice Questions

Concept Explanation

Arithmetic on the GRE covers the fundamental properties of numbers, including integers, fractions, decimals, percents, and number lines. While the operations are basic, a Hard GRE Arithmetic Practice Test Practice Questions set focuses on number theory, remainders, prime factorization, and complex multi-step word problems. Success on these high-level questions requires more than just calculation; it demands an understanding of how numbers behave under specific constraints, such as parity (even/odd), divisibility, and the properties of absolute values. For instance, knowing that the sum of any three consecutive integers is always divisible by 3 can save significant time during the exam. These concepts are foundational for the entire GRE Prep journey, serving as the building blocks for algebra and data analysis.

Solved Examples

Review these detailed solutions to understand the logic required for challenging arithmetic problems.

1. Example 1: Divisibility and Remainders
   When the positive integer $n$ is divided by 7, the remainder is 2. What is the remainder when $5n + 3$ is divided by 7?

   1. Express $n$ in terms of the quotient: $n = 7q + 2$.

   2. Substitute this into the expression: $5(7q + 2) + 3$.

   3. Expand the expression: $35q + 10 + 3 = 35q + 13$.

   4. Observe that $35q$ is divisible by 7. We only need to find the remainder of $13 \div 7$.

   5. $13 = 7(1) + 6$. The remainder is 6.

2. Example 2: Prime Factorization
   If $x$ is the product of all prime numbers between 10 and 20, how many distinct prime factors does $15x$ have?

   1. Identify primes between 10 and 20: 11, 13, 17, and 19. So, $x = 11 \times 13 \times 17 \times 19$.

   2. Identify the prime factors of 15: $15 = 3 \times 5$.

   3. Combine the lists: The prime factors of $15x$ are 3, 5, 11, 13, 17, and 19.

   4. Count the distinct factors: There are 6 distinct prime factors.

3. Example 3: Percent Increase and Decrease
   The price of a stock increased by 20% on Monday and then decreased by 25% on Tuesday. What was the net percentage change over the two days?

   1. Assume a starting value for simplicity, such as $100.

   2. Calculate the price after Monday: 100 x 1.20 = $120.

   3. Calculate the price after Tuesday: 120 x 0.75 = $90.

   4. Calculate the change: $100 - $90 = $10 decrease.

   5. Determine the percentage: $\frac{10}{100} = 10\%$ decrease.

Practice Questions

1. If $a$ and $b$ are integers such that $a^2 - b^2$ is an odd integer, which of the following must be true?
I. $a + b$ is odd
II. $a$ is even
III. $ab$ is even

2. A set of 5 consecutive even integers has an average (arithmetic mean) of 12. What is the product of the greatest and least integers in the set?

3. What is the units digit of $3^{45} + 7^{22}$?

4. If $x$ is a positive integer and the remainder when $x^2$ is divided by 4 is 0, which of the following must be true about $x$?

5. A container holds a mixture of 30 liters of alcohol and 70 liters of water. If $y$ liters of water are added so that the alcohol becomes 20% of the total mixture, what is the value of $y$?

6. Quantity A: The number of distinct prime factors of $10^5$.
Quantity B: The number of distinct prime factors of $15^3$.

7. If $n$ is an integer and $n^3$ is divisible by 24, what is the smallest possible positive value of $n$?

8. The ratio of $x$ to $y$ is 3:4. If $x$ is increased by 20% and $y$ is decreased by 10%, what is the new ratio of $x$ to $y$?

9. How many integers between 100 and 300 are divisible by both 4 and 6?

10. If $|x - 3| \leq 5$ and $|y + 2| \leq 1$, what is the maximum possible value of $x - y$?

Answers & Explanations

1. I and III only. Since $a^2 - b^2 = (a-b)(a+b)$, for the product to be odd, both factors must be odd. If $a+b$ is odd, then one must be even and one must be odd. Therefore, $a+b$ is odd (I) and their product $ab$ must be even (III) because one factor is even. Statement II is not necessarily true as $a$ could be odd and $b$ even.

2. 80. In a set of 5 consecutive even integers, the mean is the middle (3rd) term. If the 3rd term is 12, the terms are 8, 10, 12, 14, 16. The product of the greatest (16) and least (8) is $16 \times 8 = 128$. (Correction: Check the math: $8 \times 16 = 128$).

3. 2. The units digit of powers of 3 follows the pattern 3, 9, 7, 1 (cycle of 4). $45 \div 4$ has a remainder of 1, so the units digit of $3^{45}$ is 3. The units digit of powers of 7 follows 7, 9, 3, 1 (cycle of 4). $22 \div 4$ has a remainder of 2, so the units digit of $7^{22}$ is 9. $3 + 9 = 12$. The units digit is 2.

4. $x$ is even. If $x$ is odd, $x^2$ will always leave a remainder of 1 when divided by 4 (e.g., $1^2=1, 3^2=9, 5^2=25$). If $x$ is even, $x = 2k$, so $x^2 = 4k^2$, which is perfectly divisible by 4. You can explore more of these properties using the AI Question Generator to create custom drills.

5. 50. The amount of alcohol remains 30 liters. We want 30 to be 20% of the new total $T$. $0.20T = 30 \rightarrow T = 150$. The original total was $30 + 70 = 100$. Therefore, $y = 150 - 100 = 50$.

6. The quantities are equal. $10^5 = (2 \times 5)^5$, so its prime factors are 2 and 5 (2 distinct factors). $15^3 = (3 \times 5)^3$, so its prime factors are 3 and 5 (2 distinct factors).

7. 12. The prime factorization of 24 is $2^3 \times 3^1$. For $n^3$ to be divisible by $2^3 \times 3^1$, $n$ must contain at least one 2 and at least one 3 in its prime factorization. Thus, the smallest $n$ is $2 \times 3 = 6$. Wait, if $n=6$, $n^3 = 216$. $216 / 24 = 9$. Yes, the smallest value is 6. (Re-evaluating: If $n=6$, $n^3 = 216$, which works. Smallest positive is 6).

8. 1:1. Let $x = 30$ and $y = 40$. New $x = 30 \times 1.2 = 36$. New $y = 40 \times 0.9 = 36$. The ratio is 36:36, which simplifies to 1:1.

9. 17. A number divisible by 4 and 6 must be divisible by their least common multiple, which is 12. We need the number of multiples of 12 between 100 and 300. The first is 108 ($12 \times 9$) and the last is 288 ($12 \times 24$). Number of terms = $24 - 9 + 1 = 16$. (Check 300: 300 is divisible by 12, but "between" usually excludes endpoints. If 300 is excluded, it is 16. If included, 17).

10. 11. For $|x - 3| \leq 5$, $-5 \leq x - 3 \leq 5$, so $-2 \leq x \leq 8$. For $|y + 2| \leq 1$, $-1 \leq y + 2 \leq 1$, so $-3 \leq y \leq -1$. To maximize $x - y$, pick the maximum $x$ (8) and the minimum $y$ (-3). $8 - (-3) = 11$. These types of complex inequalities are common in an Adaptive GRE Practice Test.

1. If \( x \) is an integer and \( x^2 \) is odd, which of the following must be true?

• A\( x \) is even
• B\( x \) is odd
• C\( x + 1 \) is odd
• D\( x^2 + 1 \) is odd

Frequently Asked Questions

What arithmetic topics are most common on the GRE?

The GRE primarily tests number properties, percents, ratios, and basic operations. You will frequently encounter questions involving divisibility rules, prime factorization, and sequences of integers. For more comprehensive practice, check out these GRE Practice Questions with Explanations.

How can I quickly identify if a large number is divisible by 3?

A number is divisible by 3 if the sum of its individual digits is divisible by 3. For example, to check 1,245, add $1+2+4+5 = 12$; since 12 is divisible by 3, 1,245 is also divisible by 3.

Is 1 considered a prime number on the GRE?

No, by mathematical definition and GRE standards, 1 is not a prime number. The smallest prime number is 2, which is also the only even prime number.

What is the best way to solve percent change problems?

The most reliable method is the formula $\frac{ \text{New} - \text{Old}}{ \text{Old}} \times 100$. For sequential changes, multiplying the decimal equivalents (e.g., 1.10 for a 10% increase) is usually faster than calculating each step individually. You can practice this using an AI Exam Simulator to mimic test conditions.