ILTS Mathematics (208) flashcards

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

GCD-LCM product identity

For positive integers a and b, gcd(a,b) times lcm(a,b) equals a times b. This lets you find the LCM once the GCD is known, or vice versa.

Divisibility rule for 11

A number is divisible by 11 when the alternating sum of its digits, subtracting odd positions from even positions, is itself a multiple of 11, including zero.

Terminating decimal condition

A fraction in lowest terms has a terminating decimal exactly when its denominator's only prime factors are 2 and 5. Any other prime factor produces a repeating decimal.

Euclidean algorithm

To find gcd(a,b), repeatedly replace the pair with b and a mod b. The last nonzero remainder is the greatest common divisor of the original two numbers.

Modulus of a complex number

For z = a + bi, the modulus is the square root of a squared plus b squared. It equals the distance from the origin to the point (a,b).

Absolute value inequality split

The inequality |x − a| < d is equivalent to a − d < x < a + d. The inequality |x − a| > d splits into x < a − d or x > a + d.

Quadratic discriminant

For ax² + bx + c = 0, the discriminant b² − 4ac determines the roots: positive gives two real roots, zero gives one repeated root, negative gives complex conjugates.

Vertex formula for parabolas

The parabola y = ax² + bx + c has its vertex at x = −b divided by 2a. Substituting that x-value gives the maximum or minimum y-value.

Remainder theorem

Dividing a polynomial P(x) by x − r leaves a remainder equal to P(r). Consequently, x − r is a factor exactly when P(r) equals zero.

Change of base formula

The logarithm of x base b equals the natural log of x divided by the natural log of b. This lets you evaluate any base using common calculator functions.

Continuous exponential growth model

A quantity modeled by A(t) = A₀e^{kt} grows when k is positive and decays when k is negative. Doubling time equals the natural log of 2 divided by k.

Inverse function reflection

A one-to-one function and its inverse are mirror images across the line y = x. The domain of one equals the range of the other.

Geometric series sum

For a finite geometric series with first term a₁ and ratio r not equal to 1, the sum is a₁ times (1 − rⁿ) divided by (1 − r).

Horizontal asymptote by degrees

If the numerator degree is less than the denominator degree, the horizontal asymptote is y = 0. Equal degrees give the ratio of leading coefficients.

Arc length in radians

Arc length equals radius times central angle measured in radians. Sector area equals one-half radius squared times that same radian angle.

Heron's formula

Triangle area equals the square root of s(s−a)(s−b)(s−c), where s is half the perimeter and a, b, c are the side lengths.

Law of Cosines

For a triangle, c² = a² + b² − 2ab cos C. It generalizes the Pythagorean theorem to non-right triangles and finds a side from two sides and the included angle.

Similar figure scaling ratios

When two similar figures have length scale factor k, their areas scale by k² and their volumes scale by k³. Congruent figures have k equal to 1.

Sphere volume and surface area

A sphere of radius r has volume four-thirds pi r cubed and surface area four pi r squared. Surface area is the derivative of volume with respect to radius.

Polygon interior angle sum

A convex polygon with n sides has interior angles summing to (n − 2) times 180 degrees. Its exterior angles always sum to 360 degrees.

Rigid motion invariants

Translations, rotations, and reflections preserve side lengths and angle measures, producing congruent figures. Dilations change lengths by the scale factor but preserve angle measures.

Standard deviation shift rule

Adding the same constant to every data value leaves the standard deviation unchanged. Multiplying every value by k multiplies the standard deviation by the absolute value of k.

Outlier test with IQR

A value is a potential outlier if it lies below Q1 minus 1.5 times the interquartile range or above Q3 plus 1.5 times that range.

Empirical rule percentages

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

Addition rule for probability

The probability of A or B equals the probability of A plus the probability of B minus the probability of both occurring together, correcting for double counting.

Conditional probability formula

The probability of A given B equals the probability of both A and B divided by the probability of B, provided B has nonzero probability.

Permutation versus combination

Permutations count ordered arrangements using n factorial divided by (n−r) factorial. Combinations count unordered selections, dividing further by r factorial.

Expected value of a random variable

Expected value is the sum of each outcome multiplied by its probability. A game is fair when its expected net gain equals zero.

Contrapositive equivalence

The statement not Q implies not P is logically equivalent to P implies Q. The converse, Q implies P, is not equivalent and requires separate proof.

Proof by mathematical induction

Prove the claim for a base case, then assume it holds for k and show it holds for k+1. This establishes the statement for all integers beyond the base.

Counterexample role in proof

One counterexample disproves a universal statement. A single confirming example can prove an existential claim but never establishes a universal one.

Linear versus exponential model choice

Use a linear model when a quantity changes by a constant amount per unit. Use an exponential model when it changes by a constant percent per unit.