GRE Quantitative cheat sheet

Divisibility: an integer is divisible by 3 or 9 when its digit sum is; by 4 when its last two digits form a multiple of 4; by 8 when its last three digits form a multiple of 8.

If n = p^a · q^b · …, the number of positive divisors of n is (a + 1)(b + 1)…, counting every distinct prime factor.

For positive integers a and b, LCM(a, b) × GCF(a, b) = a × b; the GCF is the product of shared primes at their lowest powers.

Division algorithm: dividend = divisor × quotient + remainder, with 0 ≤ remainder < divisor; remainders can be added and multiplied.

Percent change = (new − old) ÷ old × 100%; successive changes multiply as factors, so a 20% rise followed by a 20% fall gives a net 4% decrease.

Average = sum ÷ count; a weighted average always falls between the two group averages and sits closer to the group with the larger weight.

Dividing by a fraction equals multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Exponent rules: x^a · x^b = x^(a+b); (x^a)^b = x^(ab); x^(−a) = 1/x^a for x ≠ 0; x^0 = 1 for x ≠ 0.

Combine ratios by matching the shared term: from a : b and b : c, rescale both so b is identical, then read off a : b : c.

Slope between (x1, y1) and (x2, y2) is m = (y2 − y1)/(x2 − x1); a horizontal line has slope 0 and a vertical line has undefined slope.

Line forms: y = mx + b (slope-intercept) and y − y1 = m(x − x1) (point-slope); parallel lines share m, perpendicular lines have slopes whose product is −1.

Distance between two points = √((x2 − x1)² + (y2 − y1)²); midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a for ax² + bx + c = 0; the discriminant b² − 4ac gives two real roots if positive, one if zero, none if negative.

Factoring patterns: x² + (a + b)x + ab = (x + a)(x + b); a² − b² = (a + b)(a − b); a² + 2ab + b² = (a + b)².

The parabola y = ax² + bx + c has vertex at x = −b/(2a), which is also its axis of symmetry; it opens up if a > 0 and down if a < 0.

A two-equation linear system has no solution when the lines share a slope but differ in intercept, and infinitely many solutions when both equations describe the same line.

Multiplying or dividing an inequality by a negative number reverses the symbol; |x| < k means −k < x < k, while |x| > k means x > k or x < −k.

Exponential change: after t periods, amount = P(1 + r)^t for growth and P(1 − r)^t for decay, where r is the per-period rate written as a decimal.

A triangle's interior angles sum to 180°; an exterior angle equals the sum of the two remote interior angles.

A convex n-gon has interior angles summing to (n − 2) × 180°, and its exterior angles always sum to 360°.

Pythagorean theorem: a² + b² = c² for legs a, b and hypotenuse c; common integer triples are 3-4-5, 5-12-13, 8-15-17, and 7-24-25, plus their multiples.

Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x, with the shortest side opposite the 30° angle.

Areas: triangle ½bh; parallelogram bh; trapezoid ½(b1 + b2)h; circle πr²; equilateral triangle (√3/4)s².

For a central angle θ in degrees: arc length = (θ/360) × 2πr and sector area = (θ/360) × πr²; full circumference is 2πr.

An inscribed angle is half the central angle subtending the same arc; an angle inscribed in a semicircle is 90°, and a tangent is perpendicular to the radius at the point of contact.

Similar figures with side ratio k have areas in ratio k² and volumes in ratio k³; congruent figures have k = 1.

Volumes: rectangular solid lwh; cylinder πr²h; cone (1/3)πr²h; sphere (4/3)πr³. A circle with center (h, k) and radius r satisfies (x − h)² + (y − k)² = r².

Mean = sum ÷ count and is dragged toward outliers; median is the middle value (average of the two middle values for an even count) and resists them.

Spread measures: range = max − min; IQR = Q3 − Q1 covers the middle half of the data; standard deviation is the square root of the average squared deviation from the mean.

Adding a constant to every data value shifts the mean and median by that constant but leaves the standard deviation unchanged; multiplying every value by k multiplies the standard deviation by |k|.

In a normal distribution, roughly 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Counting: arrangements of all n distinct items = n!; ordered selections of r from n = n!/(n − r)!; unordered selections = n!/(r!(n − r)!).

Probability rules: 0 ≤ P(A) ≤ 1; P(not A) = 1 − P(A); P(A or B) = P(A) + P(B) − P(A and B); P(A and B) = P(A)P(B) only when A and B are independent.

Conditional probability P(A | B) = P(A and B) ÷ P(B); two events are independent exactly when P(A | B) = P(A).

For overlapping sets, |A ∪ B| = |A| + |B| − |A ∩ B|; for three sets, subtract each pairwise overlap and add back the triple overlap.

A scatterplot's association is described by direction, form, and strength; correlation measures linear strength only and never establishes causation.