GRE Arithmetic & Number Properties: The High-Yield Rules You Must Know

The GRE is now a shorter, under-2-hour exam with two Quant sections: 12 questions in about 21 minutes, then 15 questions in about 26 minutes. The question types and math syllabus did not change, so core arithmetic and number properties remain high-yield. Because you can skip within a section and return, fast recognition of rules—and knowing when not to compute—translates directly into points.

Keep these checks at your fingertips:

• 2, 5, 10: last digit even / 0 or 5 / 0.
• 3 and 9: sum of digits divisible by 3 or 9.
• 4: last two digits form a number divisible by 4.
• 8: last three digits divisible by 8 (rare but useful for powers of 2).
• 6: divisible by both 2 and 3.
• 11: alternating-digit-sum test; if the difference is a multiple of 11 (including 0), the number is divisible by 11. Tactics: When a problem hides a large factor, strip powers of 10 (remove trailing zeros), factor out small primes early, and cancel before multiplying.

Prime facts that rescue time:

• 1 is not prime; 2 is the only even prime.
• To test if n is prime, you only need to check primes up to √n.
• Prime factorization powers everything: if n = p^a q^b r^c, then the number of positive divisors is (a+1)(b+1)(c+1).
• LCM uses highest exponents; GCF uses lowest. For positive a and b, LCM(a,b) × GCF(a,b) = a × b.
• A perfect square has an odd number of positive factors; a non-square has an even number.

Every integer n can be written as n = dq + r with 0 ≤ r < d. That remainder r is all that matters for many GRE questions.

• Add/subtract: (a ± b) mod m = [(a mod m) ± (b mod m)] mod m.
• Multiply: (ab) mod m = [(a mod m)(b mod m)] mod m.
• Powers: reduce the base first, then look for cycles (e.g., powers of 2 mod 5 cycle 2,4,3,1 and repeat).
• Negative integers: keep the remainder nonnegative. Example: −17 divided by 5 has remainder 3 because −17 = (−4)×5 + 3. Time saver: When answers are in remainder form, never compute full values—reduce early and often.

Rules that must be automatic:

• a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn). These require the same base.
• a^0 = 1 for a ≠ 0; a^(−k) = 1/a^k.
• Fractional exponents: a^(p/q) = qth-root(a^p) with a ≥ 0 if q is even.
• Square roots return the principal (nonnegative) root: √(a^2) = |a|, not a.
• Binomial trap: (a+b)^2 ≠ a^2 + b^2; it equals a^2 + 2ab + b^2.
• Units-digit cycles: powers of 2, 3, 7, 8 have length-4 cycles; 4 and 9 cycle every 2; 5 and 6 are constant.

Parity:

• even ± even = even; odd ± odd = even; odd ± even = odd.
• even × anything = even; odd × odd = odd.
• Zero is even; 0 multiplied by anything is 0. Signs:
• Product of an even number of negatives is positive; of an odd number is negative. Use parity to eliminate answer choices fast, especially in numeric entry with limited time.

• Among n consecutive integers, exactly one is divisible by n.
• Product of k consecutive integers is divisible by k! (factorial), which crushes many divisibility questions.
• Trailing zeros of n! come from pairs of 2 and 5; since 2s are abundant, count 5s: floor(n/5) + floor(n/25) + floor(n/125) + …
• Sums or averages of evenly spaced sets (arithmetic sequences) equal the average of first and last terms.

• To compare huge powers, reduce to last digits using cycles.
• For divisibility by 3 or 9, use digit sums; for 11, use alternating sums.
• When a question asks for the remainder upon division by a factor of 10 (like 2, 5), strip powers of 10 first; what remains controls the answer.

• “Integer” includes negatives and 0 unless it says “positive integer.”
• Distinct means different; consecutive means back-to-back with difference 1 (or constant step if specified).
• Don’t cancel across sums: you can cancel factors, not terms. Example: (a^2 − b^2)/(a − b) = a + b only if a ≠ b, because you factor first.
• Even roots of negatives are not real; odd roots can be negative.
• If x is an integer and x^2 is divisible by a prime p, then x is divisible by p (prime squares property).

1. Divisibility Sprint: Write 8–10 numbers and, for each, list all small divisors (2,3,4,5,6,8,9,11). Explain each rule out loud. Goal: 90 seconds per number.
2. Prime Scan: For 8 random 3-digit numbers, test primality by checking primes up to √n. Goal: under 45 seconds each.
3. Remainder Loops: Pick a modulus (7 or 9). Compute last-digit or remainder cycles for bases 2, 3, 7, 8. Then answer 6 quick questions like 7^2025 mod 9 without a calculator.
4. Exponent Laws Grid: Create 12 flash prompts mixing add/subtract/multiply exponent rules and one binomial trap each set.
5. Parity Elimination: Craft 10 expressions and predict parity and sign without computation.
6. Factor Count: Factor numbers like 72, 180, 504; list prime exponents and count divisors. Add a perfect-square example to feel the “odd number of factors” phenomenon.
7. Trailing Zeros: Compute zeros in 50!, 125!, 200! rapidly using the 5s-only method.

• Target average pace near 1 minute 44 seconds per question across Quant; build in 2–3 quick passes per section.
• First pass: harvest all rule-based freebies (parity, divisibility, last digit). Second pass: set up algebraic structure. Final pass: any remaining time for tough ones.
• Use the on-screen calculator only when it truly saves time (long division, roots of non-perfect squares, multi-digit arithmetic). Estimate before you calculate to catch key-entry errors.

• Diagnose: Start with the Quant diagnostic to surface your accuracy by subtopic (divisibility, primes, remainders, exponents/roots, odds/evens).
• Learn: Take the short lessons for any weak rule. You’ll see strategy notes and worked GRE-style examples.
• Solve Together: Walk through adaptive, step-by-step explanations on number-property questions until the patterns click.
• Test Yourself: Use 8–12 question micro-sets filtered to your weak subskills with realistic timing for each Quant section.
• Review: Exambank flags error patterns (e.g., “canceled across a sum,” “ignored absolute value,” “wrong remainder range”) and schedules spaced repetition so the rules stick.

• Divisibility rules and the 11 test.
• Prime factorization → GCF/LCM; factor-count formula.
• Remainder operations and cycle recognition.
• Exponent laws, square-root absolute-value rule, binomial expansion awareness.
• Parity and sign shortcuts.
• Trailing zeros method for n! and powers of 10.
• Plan your two-pass timing and selective calculator use.

Arithmetic and number properties convert directly into fast, low-stress points on the shorter GRE. Master the rules, rehearse them under time, and you’ll feel sections slow down for you—without doing more math than necessary.