Estimated study time: 45 minutes
Content:
GRE arithmetic covers the foundational number properties and operations tested throughout the Quantitative Reasoning sections. Unlike standardized tests for younger students, the GRE uses arithmetic in sophisticated multi-step problems and quantitative comparison questions. Mastery of arithmetic concepts is essential because they appear in questions testing algebra, data analysis, and geometry as well.
Number properties are high-yield GRE content. Integers include all whole numbers and their negatives (…, −2, −1, 0, 1, 2, …). The GRE frequently tests whether zero is positive, negative, or neither (zero is neither positive nor negative), and whether fractions and decimals are integers (they are not). Odd and even rules: odd + odd = even; even + even = even; odd + even = odd; odd × odd = odd; even × anything = even. These rules extend to exponents and are frequently used in trap questions.
Prime numbers are natural numbers greater than 1 with no positive divisors other than 1 and themselves. 2 is the only even prime. The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. GRE frequently asks about prime factorization. Every integer greater than 1 can be expressed as a unique product of prime factors (Fundamental Theorem of Arithmetic). Greatest Common Divisor (GCD) and Least Common Multiple (LCM): GCD is the largest integer dividing both numbers; LCM is the smallest positive integer divisible by both. For any two integers: GCD × LCM = Product of the two numbers.
Fractions, decimals, and percents are all representations of rational numbers, and the GRE requires fluency in converting among them and performing operations. Percent change = (New − Old) / Old × 100%. A common trap: a 50% decrease followed by a 100% increase does not return to the original value — it results in a 0% change (the 100% increase is applied to the reduced base). Successive percent changes compound: two 10% increases = 1.10 × 1.10 = 1.21 = 21% total increase, not 20%.
Ratios and proportions appear in GRE word problems constantly. If a : b = 3 : 5, you know the actual values could be 3k and 5k for any positive k. Direct proportion: y = kx. Inverse proportion: y = k/x. Rates (speed, work, concentration) are proportional relationships: distance = rate × time; work done = rate × time. Combined work rate: if worker A takes 4 hours and worker B takes 6 hours, together they complete 1/4 + 1/6 = 5/12 of the work per hour, so together they finish in 12/5 = 2.4 hours.
Absolute value, remainders, and divisibility rules are frequently tested. Divisibility: by 2 (last digit even), by 3 (digit sum divisible by 3), by 4 (last two digits divisible by 4), by 5 (ends in 0 or 5), by 9 (digit sum divisible by 9). Remainder problems: if N = 7q + 3, then N leaves remainder 3 when divided by 7. The GRE sometimes tests properties of remainders directly.
Key Terms:
Quiz Questions:
Q1. A store reduces the price of a jacket by 30%, then later increases the sale price by 30%. The final price compared to the original is:
A) The same as the original B) 9% lower than the original C) 9% higher than the original D) 30% lower than the original
Answer: B — Starting with $100: after 30% decrease = $70. After 30% increase on $70: $70 × 1.30 = $91. Final price is $9 less than $100 — a 9% decrease. Successive percent changes multiply, not add: 0.70 × 1.30 = 0.91 = 91% of original.
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Q2. Worker A can complete a project in 12 hours. Worker B can complete the same project in 8 hours. If they work together, how many hours will it take to complete the project?
A) 10 hours B) 5.6 hours C) 4.8 hours D) 20 hours
Answer: C — Combined rate = 1/12 + 1/8 = 2/24 + 3/24 = 5/24 of the job per hour. Time = 1 / (5/24) = 24/5 = 4.8 hours.
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Q3. What is the LCM of 12 and 18?
A) 6 B) 36 C) 216 D) 72
Answer: B — Prime factorize: 12 = 2² × 3; 18 = 2 × 3². LCM takes the highest power of each prime: 2² × 3² = 4 × 9 = 36. Check: GCD(12,18) = 6; GCD × LCM = 12 × 18 = 216 = 6 × 36. Confirmed.
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Q4. When integer N is divided by 7, the remainder is 4. When N is divided by 3, the remainder is 1. Which of the following could be N?
A) 22 B) 25 C) 18 D) 32
Answer: B — Test each: 25 ÷ 7 = 3 remainder 4 (check); 25 ÷ 3 = 8 remainder 1 (check). Answer is 25. Check 22: 22 ÷ 7 = 3 r 1 (fails first condition). Check 18: 18 ÷ 7 = 2 r 4 (check); 18 ÷ 3 = 6 r 0 (fails). Check 32: 32 ÷ 7 = 4 r 4 (check); 32 ÷ 3 = 10 r 2 (fails).
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Q5. The sum of three consecutive even integers is 78. What is the largest of the three integers?
A) 24 B) 26 C) 28 D) 30
Answer: C — Let the integers be n, n+2, n+4. Sum: 3n + 6 = 78 → 3n = 72 → n = 24. The integers are 24, 26, 28. Largest = 28.
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