Divisibility rule for 3
Answer: A number is divisible by 3 if the sum of its digits is divisible by 3.
Explanation
Example: 123 → 1+2+3=6, divisible by 3, so 123 is divisible by 3.
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GRE Quantitative: Arithmetic, Algebra, Geometry and Data Formulas (University)
Divisibility & RemaindersEven, Odd & PrimePercentsExponents & RootsAlgebraic IdentitiesTrianglesCirclesCountingProbabilityStatisticsQC Traps
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Divisibility & Remainders
Divisibility rule for 4
Answer: A number is divisible by 4 if its last two digits form a number divisible by 4.
Explanation
Example: 7,316 → last two digits 16, 16/4=4, so 7,316 is divisible by 4.
Divisibility rule for 9
Answer: A number is divisible by 9 if the sum of its digits is divisible by 9.
Explanation
Example: 7,281 → 7+2+8+1=18, divisible by 9, so 7,281 is divisible by 9.
Divisibility rule for 6
Answer: A number is divisible by 6 if it is divisible by both 2 and 3.
Explanation
Example: 72 is even and 7+2=9 (divisible by 3), so 72 is divisible by 6.
Remainder formula
Answer: Dividend = Divisor × Quotient + Remainder, where 0 ≤ Remainder < Divisor.
Explanation
Example: 17 ÷ 5 → 17 = 5×3 + 2, so quotient 3, remainder 2.
Even, Odd & Prime
Sum rules for even/odd numbers
Answer: Even+Even=Even, Odd+Odd=Even, Even+Odd=Odd.
Explanation
Example: 4+6=10 (even), 3+5=8 (even), 4+3=7 (odd).
Product rules for even/odd numbers
Answer: Even×Even=Even, Odd×Odd=Odd, Even×Odd=Even.
Explanation
Example: 6×4=24 (even), 3×5=15 (odd), 6×5=30 (even).
What defines a prime number, and what's unique about 2?
Answer: A prime has exactly two positive divisors: 1 and itself. 2 is the only even prime; all other primes are odd.
Explanation
Example: 2, 3, 5, 7, 11 are prime; 2 is even, the rest are odd.
Percents
Percent change formula
Answer: Percent change = (New − Old)/Old × 100%.
Explanation
Example: price rises from $50 to $65 → (65−50)/50×100% = 30% increase.
Why don't successive percent changes add directly?
Answer: Each percent applies to the new (already changed) value, so +20% then −20% does NOT return to the original amount; multiply the factors instead.
Explanation
Example: 100 → +20% = 120 → −20% of 120 = 96, not 100.
How do you find 'x% of y% of a number'?
Answer: Multiply the decimal forms successively: Number × (x/100) × (y/100).
Explanation
Example: 20% of 50% of 200 = 200×0.20×0.50 = 20.
Exponents & Roots
Exponent rule: multiplying same base
Answer: a^m × a^n = a^(m+n).
Explanation
Example: 2^3 × 2^4 = 2^7 = 128.
Exponent rule: power of a power
Answer: (a^m)^n = a^(m×n).
Explanation
Example: (3^2)^4 = 3^8.
Negative exponent rule
Answer: a^(−n) = 1/a^n (for a ≠ 0).
Explanation
Example: 2^(−3) = 1/2^3 = 1/8.
Fractional exponent rule
Answer: a^(1/n) = the nth root of a; a^(m/n) = (nth root of a)^m.
Explanation
Example: 8^(1/3) = cube root of 8 = 2; 8^(2/3) = 2^2 = 4.
Root multiplication/division rule
Answer: √a × √b = √(ab); √a / √b = √(a/b), for a, b ≥ 0.
Explanation
Example: √2 × √8 = √16 = 4.
Algebraic Identities
Difference of squares identity
Answer: a² − b² = (a − b)(a + b).
Explanation
Example: 49 − 4 = 7² − 2² = (7−2)(7+2) = 5×9 = 45.
Perfect square identities
Answer: (a+b)² = a² + 2ab + b²; (a−b)² = a² − 2ab + b².
Explanation
Example: (x+3)² = x²+6x+9; (x−3)² = x²−6x+9.
Triangles
45-45-90 triangle side ratio
Answer: Legs : Legs : Hypotenuse = x : x : x√2.
Explanation
Example: legs of 5 give a hypotenuse of 5√2 ≈ 7.07.
30-60-90 triangle side ratio
Answer: Side opposite 30° : opposite 60° : hypotenuse = x : x√3 : 2x.
Explanation
Example: if x=4, sides are 4, 4√3, and 8.
Triangle inequality theorem
Answer: The sum of any two sides of a triangle must be greater than the third side.
Explanation
Example: sides 4 and 7 mean the third side must be between 3 and 11.
Circles
Arc length formula
Answer: Arc length = (central angle/360°) × 2πr.
Explanation
Example: a 90° arc on a circle of radius 4 = (90/360)×2π(4) = 2π.
Sector area formula
Answer: Sector area = (central angle/360°) × πr².
Explanation
Example: a 60° sector of radius 6 = (60/360)×π(36) = 6π.
Inscribed angle theorem
Answer: An inscribed angle is half the measure of the central angle that subtends the same arc.
Explanation
Example: if a central angle is 80°, the inscribed angle on the same arc is 40°.
Counting
Permutations formula
Answer: nPr = n! / (n − r)!, the number of ordered arrangements of r items from n.
Explanation
Example: arranging 3 of 5 people in order: 5P3 = 5!/2! = 60.
Combinations formula
Answer: nCr = n! / [r!(n − r)!], the number of unordered selections of r items from n.
Explanation
Example: choosing 3 of 5 people: 5C3 = 5!/(3!2!) = 10.
Permutations vs. combinations — key distinction
Answer: Use permutations when order matters (e.g., rankings); use combinations when order doesn't matter (e.g., groups or committees).
Explanation
Example: arranging 1st/2nd/3rd place is a permutation; picking 3 team members is a combination.
Probability
Probability of independent events (A and B)
Answer: P(A and B) = P(A) × P(B) when events are independent.
Explanation
Example: two fair coin flips both heads: 1/2 × 1/2 = 1/4.
Probability of mutually exclusive events (A or B)
Answer: P(A or B) = P(A) + P(B) when events cannot both occur.
Explanation
Example: rolling a 2 or a 5 on a die: 1/6 + 1/6 = 1/3.
Statistics
Mean (average) formula
Answer: Mean = (sum of all values) / (number of values).
Explanation
Example: mean of 2, 4, 6 = (2+4+6)/3 = 4.
Median definition
Answer: The middle value in an ordered data set (average of the two middle values if the count is even).
Explanation
Example: median of 1, 3, 7 is 3; median of 1, 2, 3, 4 is (2+3)/2 = 2.5.
What does standard deviation measure?
Answer: The typical distance of data points from the mean — a larger SD means more spread-out data.
Explanation
Example: {10,10,10} has SD 0; {1,10,19} has a large SD despite sharing the same mean as {10,10,10}.
Normal distribution: 68-95-99.7 rule
Answer: About 68% of data falls within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
Explanation
Example: if mean=100 and SD=10, about 68% of values lie between 90 and 110.
QC Traps
QC trap #1: Zero
Answer: Always test whether a variable could be zero — it can change inequalities, multiplication, or division results unexpectedly.
Explanation
Example: if x²>x is claimed, it's false when x=0.
QC trap #2: Negative numbers
Answer: Test negative values for variables — they can flip inequality direction when multiplying/dividing or change the sign of results.
Explanation
Example: if x<0, then x² > x, unlike when x is positive.
QC trap #3: Fractions between 0 and 1
Answer: Fractions between 0 and 1 behave oddly: squaring makes them smaller, and their reciprocal is greater than 1.
Explanation
Example: if x=1/2, x² = 1/4 < x, opposite of what happens when x>1.
QC trap #4: Testing only positive integers
Answer: Don't assume a variable is a positive integer unless stated — test 0, negatives, and fractions too, since they often change the answer.
Explanation
Example: 'x² > 4' seems to imply x>2, but x could be −3, which also satisfies it.
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