GRE Quantitative flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Least common multiple

The smallest positive integer divisible by each of two or more integers. For two positive integers a and b, LCM(a, b) = a × b ÷ GCF(a, b).

Greatest common factor

The largest integer that divides two or more integers with no remainder. It equals the product of all shared prime factors, each taken to its lowest power.

Divisibility rule for 3

An integer is divisible by 3 exactly when the sum of its digits is divisible by 3. The identical digit-sum test determines divisibility by 9.

Counting positive divisors

If n = p^a × q^b in prime factorization, then n has (a + 1)(b + 1) positive divisors. Add one to each exponent and multiply the results.

Successive percent changes

Consecutive percent changes multiply as factors rather than adding. A 20% increase followed by a 20% decrease gives 1.2 × 0.8 = 0.96, a net 4% loss.

Weighted average

The sum of each value multiplied by its weight, divided by the total weight. The result always lies closer to the value carrying the larger weight.

Remainder relationship

For positive integers, dividend = divisor × quotient + remainder, where the remainder is at least 0 and strictly less than the divisor.

Dividing by a fraction

Dividing by a fraction is equivalent to multiplying by its reciprocal. Thus dividing by 3/4 is the same as multiplying by 4/3.

Slope formula

For points (x1, y1) and (x2, y2), slope m = (y2 − y1) ÷ (x2 − x1). It measures vertical change per unit of horizontal change.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals of each other, so the product of the slopes equals −1.

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any quadratic, including ones that do not factor.

Discriminant

For ax² + bx + c = 0, the discriminant is b² − 4ac. A positive value gives two real roots, zero gives one repeated root, and negative gives none.

Difference of squares

The expression a² − b² factors as (a + b)(a − b). For example, x² − 49 factors as (x + 7)(x − 7).

Vertex of a parabola

The parabola y = ax² + bx + c has its vertex at x = −b ÷ (2a). The vertex is a minimum when a > 0 and a maximum when a < 0.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number flips the inequality symbol, so −2x < 6 becomes x > −3.

Parallel lines in systems

A system of two linear equations has no solution when the graphs have the same slope but different y-intercepts, since the lines never meet.

Polygon interior angle sum

A convex polygon with n sides has interior angles summing to (n − 2) × 180°. A quadrilateral therefore has interior angles totaling 360°.

Pythagorean triples

Whole-number right triangles include 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Any integer multiple of a triple is also a triple.

45-45-90 triangle

An isosceles right triangle has equal legs of length x and hypotenuse x√2. Its side ratio is 1 : 1 : √2.

30-60-90 triangle

Its side ratio is 1 : √3 : 2. The shortest side faces the 30° angle, and the hypotenuse is twice that shortest side.

Circle area and circumference

A circle of radius r has area πr² and circumference 2πr. Since diameter d = 2r, circumference also equals πd.

Arc length and sector area

For a central angle θ in degrees, arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².

Inscribed angle theorem

An inscribed angle measures half the central angle that subtends the same arc. An angle inscribed in a semicircle is therefore 90°.

Similar figure ratios

When two similar figures have corresponding side lengths in ratio k, their areas are in ratio k² and their volumes in ratio k³.

Mean versus median

The mean is the sum divided by the count and is pulled toward outliers. The median is the middle value and is resistant to extreme values.

Standard deviation

The square root of the average squared deviation from the mean. It measures spread, and adding a constant to every data value leaves it unchanged.

Interquartile range

IQR = Q3 − Q1, the spread of the middle half of a data set. It ignores the lowest and highest quarters, so outliers do not affect it.

Combinations formula

The number of unordered selections of r objects from n distinct objects is n! ÷ (r!(n − r)!). Order does not matter in a combination.

Permutations formula

The number of ordered arrangements of r objects chosen from n distinct objects is n! ÷ (n − r)!. Order matters in a permutation.

Independent events probability

For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B) with no overlap.

General addition rule

P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes outcomes counted twice because they belong to both events.

Normal distribution percentages

In a normal distribution about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.