ILTS Mathematics (208) cheat sheet

Divisibility: 3 iff digit sum divisible by 3; 4 iff last two digits form a multiple of 4; 8 iff last three digits form a multiple of 8; 9 iff digit sum divisible by 9; 11 iff alternating digit sum is a multiple of 11.

GCD-LCM identity: for positive integers a and b, gcd(a,b) · lcm(a,b) = a · b, so lcm(a,b) = ab / gcd(a,b).

Euclidean algorithm: gcd(a,b) = gcd(b, a mod b); repeat until remainder is 0, then the last nonzero divisor is the gcd.

Rational numbers have terminating decimal expansions exactly when the reduced denominator has only 2 and 5 as prime factors; otherwise the expansion repeats.

Irrationality proof pattern: assume √n = p/q in lowest terms, derive p² = nq², force p and q both divisible by a prime factor of n, contradicting lowest terms.

Complex arithmetic: (a+bi)(c+di) = (ac−bd) + (ad+bc)i; modulus |a+bi| = √(a²+b²); conjugate product (a+bi)(a−bi) = a²+b².

Absolute value as distance: |x − a| < d means a − d < x < a + d; |x − a| > d splits into x < a − d or x > a + d.

Percent change = (new − old)/old × 100; successive percent changes of p% then q% multiply factors (1 + p/100)(1 + q/100), not add.

Scientific notation rules: a × 10ⁿ with 1 ≤ |a| < 10; multiplication adds exponents, division subtracts, and raising to a power multiplies the exponent.

Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a); discriminant b² − 4ac > 0 gives two real roots, = 0 one repeated root, < 0 complex conjugate roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); vertex form y = a(x − h)² + k has vertex (h, k) and axis x = h.

Factor/remainder theorems: (x − r) is a factor of P(x) exactly when P(r) = 0; the remainder on dividing by (x − r) is P(r).

Logarithm laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xᵏ) = k log x; change of base log_b x = ln x / ln b.

Exponential growth/decay: A(t) = A₀(1 + r)ᵗ for discrete growth, A(t) = A₀e^{kt} for continuous; doubling time = ln 2 / k.

Function transformations: f(x − h) shifts right h, f(x) + k shifts up k, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, af(x) stretches vertically by |a|.

Inverse functions: f⁻¹ exists only if f is one-to-one; graphs of f and f⁻¹ are mirror images across y = x; domain and range swap.

Arithmetic sequence aₙ = a₁ + (n − 1)d with sum Sₙ = n(a₁ + aₙ)/2; geometric aₙ = a₁rⁿ⁻¹ with sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1.

Rational function asymptotes: vertical asymptotes at zeros of the denominator (after canceling common factors); horizontal asymptote determined by comparing numerator and denominator degrees.

Circle measures: circumference C = 2πr, area A = πr²; arc length = rθ for central angle θ in radians; sector area = ½r²θ.

Triangle area = ½bh; Heron's formula A = √[s(s−a)(s−b)(s−c)] with s = (a+b+c)/2; Law of Sines a/sin A = b/sin B = c/sin C.

Law of Cosines c² = a² + b² − 2ab cos C; when C = 90° it reduces to the Pythagorean theorem.

Similar figures: corresponding lengths scale by k, areas by k², volumes by k³; congruent figures have scale factor 1.

Volume formulas: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³ with surface area 4πr².

Coordinate geometry: distance = √[(x₂−x₁)² + (y₂−y₁)²]; midpoint = ((x₁+x₂)/2, (y₁+y₂)/2); slope m = (y₂−y₁)/(x₂−x₁).

Parallel lines cut by a transversal: alternate interior and corresponding angles are congruent; consecutive interior angles are supplementary.

Polygon angle sums: interior sum = (n − 2)·180°; exterior sum = 360° for any convex polygon.

Rigid motions (translation, rotation, reflection) preserve distance and angle measure; dilations preserve shape but scale length by the factor k.

Mean = sum/n; median is the middle value of ordered data (average of two middle values when n is even); mode is the most frequent value.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves SD unchanged, multiplying by k multiplies SD by |k|.

Outliers in a box plot are values beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR, where IQR = Q3 − Q1.

Normal distribution: about 68% of data lie within 1 SD of the mean, 95% within 2 SD, 99.7% within 3 SD; z = (x − μ)/σ.

Probability rules: P(Aᶜ) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); independent events satisfy P(A ∩ B) = P(A)·P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B); mutually exclusive events have P(A ∩ B) = 0.

Counting: permutations nPr = n!/(n−r)! order matters; combinations nCr = n!/[r!(n−r)!] order does not; with repetition, nʳ outcomes for r slots.

Expected value E(X) = Σ x·P(x); a fair game has expected net gain of zero, and the mean of a large sample tends toward E(X) by the Law of Large Numbers.

Correlation r ranges from −1 to 1 and measures strength of linear association only; correlation does not establish causation.

Direct proof: assume the hypothesis, apply definitions and known theorems, conclude the claim; contrapositive proof proves ¬Q → ¬P, which is logically equivalent to P → Q.

Proof by contradiction assumes the negation of the statement and derives an impossibility; proof by induction verifies a base case and the step P(k) → P(k+1).

Counterexample disproves a universal claim; a single example never proves a universal claim but can prove an existential one.

Converse of P → Q is Q → P and is not logically equivalent; the contrapositive ¬Q → ¬P is equivalent.

Modeling cycle: identify quantities and units, build a function or equation, solve, interpret in context, and check reasonableness of the result against the original situation.

Linear model y = mx + b has constant rate of change m; exponential model y = abˣ has constant percent change, so use exponential when growth is multiplicative.

Unit analysis: multiply by conversion ratios so unwanted units cancel; dimensional consistency requires both sides of an equation to share units.

Rounding and precision: report results to the least precise measurement; percent error = |measured − accepted| / |accepted| × 100.

Function as model: domain restrictions come from context, such as nonnegative time, integer counts, and capacity limits on a container.