Pancyprian Core Math cheat sheet

Divisibility: 3 and 9 by digit sum; 4 by last two digits; 8 by last three digits; 11 by alternating digit sums differing by a multiple of 11.

For positive integers a, b: gcd(a,b)·lcm(a,b) = a·b; use prime factorisation with minimum exponents for gcd and maximum exponents for lcm.

Laws of exponents: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(m/n) = the n-th root of a^m for a > 0.

Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^k) = k log x, log_b b = 1, log_b 1 = 0, change of base log_b x = ln x / ln b.

Percent change = (new − old)/old × 100%; successive changes of p% then q% give a factor (1 + p/100)(1 + q/100), not p + q.

Compound interest A = P(1 + r/n)^(nt); continuous growth A = Pe^(rt); simple interest A = P(1 + rt).

Rationalise denominators by multiplying by the conjugate: 1/(√a + √b) = (√a − √b)/(a − b).

|x| < k means −k < x < k; |x| > k means x < −k or x > k, for k > 0.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 one repeated root, < 0 no real roots.

Vertex of y = ax² + bx + c is at x = −b/(2a); sum of roots = −b/a and product = c/a.

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

Exponential y = a·b^x grows for b > 1 and decays for 0 < b < 1; logarithmic y = log_b x is its inverse, with domain x > 0.

Transformations of y = f(x): f(x) + k shifts up by k, f(x + k) shifts left by k, −f(x) reflects in the x-axis, f(−x) reflects in the y-axis, a·f(x) scales vertically by a.

Domain restrictions: exclude values making a denominator zero and require the radicand of an even root to be non-negative.

Arithmetic sequence a_n = a_1 + (n − 1)d, S_n = n(a_1 + a_n)/2; geometric a_n = a_1 r^(n−1), S_n = a_1(1 − r^n)/(1 − r) for r ≠ 1.

Solve simultaneous linear equations by elimination or substitution; the system has no unique solution when the coefficient determinant ad − bc = 0.

Circle: circumference 2πr, area πr²; arc length = rθ and sector area = ½r²θ only when θ is in radians.

Triangle area = ½bh = ½ab sin C; Pythagorean theorem a² + b² = c² holds only for right triangles.

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

Similar figures: corresponding lengths scale by k, areas by k², volumes by k³.

Interior angle sum of an n-gon is (n − 2)·180°; each interior angle of a regular n-gon is (n − 2)·180°/n.

Distance √((x₂−x₁)² + (y₂−y₁)²); midpoint ((x₁+x₂)/2, (y₁+y₂)/2); gradient (y₂−y₁)/(x₂−x₁).

Parallel lines have equal gradients; perpendicular gradients multiply to −1.

Equation of a circle centred at (h, k) with radius r: (x − h)² + (y − k)² = r².

Convert degrees to radians by multiplying by π/180; 180° = π radians, so arc and sector formulas require radians.

Identities: sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, 1 + tan²θ = sec²θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2, cos 60° = ½.

Sine rule: a/sin A = b/sin B = c/sin C = 2R; use it with a known side opposite a known angle.

Cosine rule: a² = b² + c² − 2bc cos A; use it for SAS or SSS, and it reduces to Pythagoras when A = 90°.

Sine is positive in quadrants I and II, cosine in I and IV, tangent in I and III; use a reference angle to find all solutions in a given interval.

Graphs: y = sin x and y = cos x have period 360° (2π) and amplitude 1; y = tan x has period 180° (π) with asymptotes at 90° + 180°k.

For y = A sin(B(x − C)) + D the amplitude is |A|, period 360°/|B| (or 2π/|B|), phase shift C and vertical shift D.

Mean = Σx/n; for grouped data use class midpoints; median is the middle value of ordered data; mode is the most frequent value.

Sample variance s² = Σ(x − x̄)²/(n − 1); population variance uses n; standard deviation is the positive square root of variance.

Adding a constant c to every value adds c to the mean and median but leaves variance, standard deviation and range unchanged.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0.

P(A | B) = P(A ∩ B)/P(B); events are independent when P(A ∩ B) = P(A)·P(B).

Permutations nPr = n!/(n − r)!; combinations nCr = n!/(r!(n − r)!); use combinations when order does not matter.

Binomial distribution P(X = k) = nCk·p^k·(1 − p)^(n−k), with mean np and variance np(1 − p).

For a normal distribution about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

Derivative definition: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; it gives the gradient of the tangent and the instantaneous rate of change.

Power rule d/dx (x^n) = n·x^(n−1); also d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (e^x) = e^x, d/dx (ln x) = 1/x.

Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v²; chain rule dy/dx = dy/du · du/dx.

Stationary points occur where f′(x) = 0; f″(x) > 0 gives a local minimum, f″(x) < 0 a local maximum.

Indefinite integral ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C.

Definite integral ∫ from a to b of f(x) dx equals the signed area between the curve and the x-axis; areas below the axis count as negative.

Fundamental theorem: if F′ = f then ∫ from a to b of f(x) dx = F(b) − F(a).

Area between two curves is ∫(upper − lower) dx over the interval between their intersection points.

Direct proof: assume the hypothesis and derive the conclusion by valid steps; a single counterexample disproves a universal statement.

Proof by contradiction assumes the negation of the claim and derives a falsehood; proof by induction needs a base case plus the step from n to n + 1.

Converse of p → q is q → p (not equivalent); contrapositive is ¬q → ¬p (equivalent); inverse is ¬p → ¬q (not equivalent).

Necessary condition: q is necessary for p when p → q; sufficient condition: p is sufficient for q when p → q.

Pigeonhole principle: placing n + 1 objects into n boxes forces at least one box to hold two or more objects.

Parity argument: the sum of two odd numbers is even, odd × odd is odd, and no integer square is congruent to 2 or 3 modulo 4.

Check a solution by substituting back into the original equation, and reject roots introduced by squaring or by multiplying by an expression that can be zero.

Estimate before computing: round to one significant figure to bound the answer, and keep units consistent throughout a multi-step problem.