Mathematics
Divisibility by 11 — Alternating sum of digits must be divisible by 11. For 2728: (2+2)−(7+8)=−11, divisible. Quick check for factorization problems.
Sum of first n naturals — n(n+1)/2. Remember: 100 naturals sum to 100×101/2=5050. Useful anchor for series problems.
Pythagorean triplets generator — For odd n: (n, (n²−1)/2, (n²+1)/2). Example: n=5 gives (5,12,13). For even n=2m: (2m, m²−1, m²+1).
AM≥GM application — For positive a,b: (a+b)/2 ≥ √(ab), equality when a=b. To minimize/maximize products with fixed sums, use this inequality directly.
Remainder when dividing by 9 — Same as digit sum's remainder. 7823 mod 9 = (7+8+2+3) mod 9 = 20 mod 9 = 2. Instant verification for calculations.
Quadratic discriminant sign — b²−4ac: positive=two roots, zero=one root, negative=no real roots. Remember "Positive→Two, Zero→One, Negative→None."
Combinatorial identity nCr — nCr = nC(n−r); also nCr + nC(r−1) = (n+1)Cr. Use symmetry to reduce calculations when r > n/2.
Triangle inequality shortcut — In triangle with sides a,b,c: each side < sum of other two. Quick elimination: if largest side ≥ sum of others, not a triangle.
Geometric series sum — a(rⁿ−1)/(r−1) for r≠1. Remember infinite sum a/(1−r) only when |r|<1. Common trap in convergence problems.
Binomial coefficient peaks — In (a+b)ⁿ, middle term(s) have largest coefficient. For even n: term at n/2; for odd n: two middle terms equal.
Polynomial remainder theorem — P(x) divided by (x−a) gives remainder P(a). Instant evaluation without long division; factor theorem when P(a)=0.
Cyclicity of unit digits — Powers of 2,3,7,8 cycle every 4; 4,9 every 2; 5,6 stay same; 1 always 1; 0 always 0. Find last digit of large powers quickly.
Logarithm base change — log_b(a) = log(a)/log(b). When bases differ, convert all to common base (usually 10 or e) for calculation.
Distance formula memory — √[(x₂−x₁)²+(y₂−y₁)²]. Think "run² + rise²" under root. Section formula: weighted average of coordinates.
Coordinate geometry slope — Parallel lines: equal slopes (m₁=m₂). Perpendicular lines: m₁×m₂=−1. Product of perpendicular slopes always negative one.