Arithmetic

Last updated July 07, 2026 · Reviewed by the AnvayaPrep team

Introduction

Arithmetic is the foundation of all GRE Quantitative Reasoning. The unit spans 37 topics covering every arithmetic concept tested on the exam: integers and number properties (even, odd, prime, composite, factors, multiples, divisibility), fractions and operations, decimals and conversions, percentages (basics, percent change, percent increase and decrease, percent comparison), ratios and proportions, number lines, absolute values, remainders, scientific notation, and estimation strategies. Arithmetic topics appear in every GRE question format -- straightforward calculations, quantitative comparisons, data interpretation, and word problems -- and they underlie every other quantitative topic in the exam.

The GRE does not test advanced arithmetic. It tests whether students can apply arithmetic concepts accurately under time pressure, recognize patterns in number properties, and avoid the specific traps that the test consistently sets. Students who understand why arithmetic rules work -- not just that they work -- solve problems faster and make fewer careless errors under exam conditions.

Learning Objectives

• Apply the four arithmetic operations fluently to integers, fractions, decimals, and mixed numbers without computational errors
• Distinguish between even, odd, prime, composite, and special numbers (0, 1, negative integers) and predict how they behave under operations
• Use prime factorization to compute greatest common factors, least common multiples, and to simplify fractions efficiently
• Convert fluently between fractions, decimals, and percentages in both directions
• Calculate percent change using the correct reference value (the original, not the new value), and compute net effects of successive percent changes using multiplication rather than addition
• Set up and solve ratio and proportion problems, distinguishing part-to-part ratios from part-to-whole ratios
• Apply divisibility rules to quickly identify factors without long division
• Use estimation and approximation strategically to check reasonableness of computed answers and to solve comparison problems efficiently
• Interpret and perform operations on numbers expressed in scientific notation
• Solve problems involving absolute value, including absolute value equations and inequalities

High-Yield Concepts

Number Properties and Integer Behavior

Number properties form the analytical backbone of GRE arithmetic. The most heavily tested concepts involve even/odd behavior under operations, prime numbers and their unique properties, and divisibility.

Even and odd rules are exact and exploitable. When a product of two integers is odd, both integers must be odd -- this single deduction can eliminate multiple answer choices instantly. When a sum of two integers is odd, exactly one is even and one is odd.

Prime numbers: 2 is the only even prime. 1 is neither prime nor composite. The primes below 20 are 2, 3, 5, 7, 11, 13, 17, 19 -- these appear constantly in GRE problems. Prime factorization underlies GCF and LCM calculations and divisibility analysis.

Critical special cases: 0 is even. Negative integers can be even or odd. These edge cases are frequent GRE traps in problems that state "for all integers n" without restricting to positive integers.

Fractions, Decimals, and Percentage Conversions

Fluent conversion between forms is required for nearly every arithmetic question. The fundamental relationship: percent means "per hundred," so 45% = 45/100 = 0.45. The conversion chain (percent to decimal: divide by 100; decimal to percent: multiply by 100; fraction to decimal: divide numerator by denominator) must be automatic.

The most frequently tested computation error is adding numerators and denominators directly when adding fractions. This is incorrect: 1/2 + 1/3 = 5/6, not 2/5. Fractions require a common denominator for addition and subtraction; only multiplication and division work straight across.

Percent change requires dividing by the original value, not the new value. This is the most common percent error on the GRE. If a quantity increases from 80 to 100, the percent change is (100-80)/80 = 25%, not (100-80)/100 = 20%. The reference point is always the starting value.

Successive percent changes multiply, they do not add. A 20% increase followed by a 20% decrease does not return to the original: 1.2 x 0.8 = 0.96, a net decrease of 4%. Students who add the changes get the wrong answer.

Ratios and Proportional Reasoning

A ratio compares two quantities. Part-to-part ratios (boys to girls: 3:2) differ from part-to-whole ratios (boys to all students: 3/5). Confusing them is a common trap. Given a part-to-part ratio of 3:2, the part-to-whole ratios are 3/5 and 2/5.

Ratios alone do not determine actual values. If the ratio of apples to oranges is 3:4, there could be 3 and 4, 6 and 8, or 300 and 400. An additional constraint (like a total or a specific quantity) is required to solve for actual values. GRE problems frequently provide this constraint in the question or answer choices.

Proportions use cross-multiplication: if a/b = c/d, then ad = bc. This works for unit conversions, scale problems, and mixture problems alike.

Study Strategy

Begin with number properties, prime numbers, and divisibility rules. These topics unlock efficient reasoning on dozens of other problems and are prerequisites for fractions, LCM, and GCF calculations.

Then master fractions and decimals as a paired unit -- practice converting between forms until the conversions are automatic. Follow with percentages (basics, then percent change, then successive changes). These build sequentially and the later topics become straightforward once the earlier ones are internalized.

Study ratios and proportions after percentages since they use the same proportional reasoning framework. Then tackle remaining specialized topics (absolute value, scientific notation, remainders, estimation) individually.

Practice arithmetic calculations without a calculator. The GRE Quantitative section provides an on-screen calculator, but it is slower than mental math for simple operations. Students who rely on the calculator for basic fraction and percent calculations lose significant time.

Common Mistakes

Dividing by the wrong value in percent change. Always divide by the original (starting) value, never the new (ending) value. This error appears in questions about price changes, population changes, and salary adjustments.

Adding percent changes instead of multiplying. Successive percent changes compound multiplicatively. A 30% increase followed by a 30% decrease does not return to the original; the multipliers are 1.3 and 0.7, giving a net of 0.91 -- a 9% decrease.

Treating 1 as prime. One has exactly one factor (itself), so it fails the definition of a prime number (which requires exactly two distinct factors). Counting 1 as prime will cause errors on questions about prime counts and prime factorization.

Forgetting that 0 is even and can be a valid value for integer variables. GRE problems that say "n is an integer" allow n = 0 unless explicitly excluded. Testing n = 0 can often disprove an answer choice that seems to always hold.

Confusing part-to-part and part-to-whole ratios. If A:B = 3:4, then A represents 3/7 of the total (3 + 4 = 7), not 3/4.

Adding numerators and denominators directly when adding fractions. 1/3 + 1/4 is 7/12, not 2/7. Always find a common denominator before adding or subtracting.

Exam Tips

Use estimation aggressively to check whether computed answers are reasonable before selecting. If a percent change calculation gives you 340%, double-check -- most GRE percent change answers fall between -100% and +200%.

When a question asks about "all integers" or "for any integer n," always test the special cases: n = 0, n = 1, n = -1, and n = 2. These edge cases defeat many seemingly universal statements.

For divisibility questions, memorize the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10. These allow you to factor large numbers without long division and are faster than using the calculator.

On ratio questions, express the ratio using a multiplier variable (if ratio is 3:4, let quantities be 3k and 4k) and solve for k using the additional constraint provided.