Mathematics
Divisibility by 11 — Alternating sum of digits must be divisible by 11. For 2728: 2−7+2−8 = −11, divisible by 11. Quick check for factorization problems.
Sum of first n natural numbers — n(n+1)/2. Remember: 1+2+3+…+100 = 100×101/2 = 5050. Memorize this anchor example for instant pattern recognition.
Perfect squares ending digits — Perfect squares can only end in 0,1,4,5,6,9. Never 2,3,7,8. Eliminates wrong options instantly in multiple-choice questions.
Arithmetic-Geometric mean inequality — AM ≥ GM: (a+b)/2 ≥ √(ab). Equality when a=b. Essential for optimization problems and finding maximum/minimum values.
Integration by parts mnemonic (ILATE) — Choose u in order: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. First function in list becomes u in ∫u dv = uv − ∫v du.
Quadratic formula sign check — For ax²+bx+c=0, sum of roots = −b/a, product = c/a. Quick verification: if roots are 2,3 then sum=5, product=6 means equation is x²−5x+6=0.
Binomial theorem last digit — For (a+b)ⁿ, use modular arithmetic. Example: last digit of 7¹⁵ = last digit of 7⁴×3+3 = (last digit of 2401)³×7 = 1×7 = 7.
Euler's totient for prime powers — φ(pⁿ) = pⁿ − pⁿ⁻¹ = pⁿ⁻¹(p−1). For p prime, φ(p)=p−1. Crucial for number theory problems in CMI.
Triangle inequality shortcut — For sides a,b,c: |a−b| < c < a+b. If this fails, triangle impossible. Saves time in geometry validity checks.
Pigeonhole principle quick form — If n+1 objects in n boxes, at least one box has ≥2 objects. Generalized: ⌈(objects)/(boxes)⌉ in some box. Core technique for CMI combinatorics.
Geometric series sum — a + ar + ar² + … + arⁿ⁻¹ = a(rⁿ−1)/(r−1). For infinite series when |r|<1: a/(1−r). Memorize both forms.
Fermat's Little Theorem — If p is prime and a not divisible by p, then aᵖ⁻¹ ≡ 1 (mod p). Corollary: aᵖ ≡ a (mod p). Essential for modular arithmetic.
Computer Science
Time complexity hierarchy — O(1) < O(log n) < O(n) < O(n log n) < O(n²) < O(n³) < O(2ⁿ) < O(n!). Memorize this order for algorithm comparison.
Stack vs Queue mnemonic — Stack = LIFO (Last In First Out) like plates stacked; Queue = FIFO (First In First Out) like ticket line. Function calls use stack.
Binary search condition — Array must be sorted. Works only on sorted data. Time: O(log n). Common mistake: applying to unsorted arrays yields wrong results.
Tree traversal order (DIP) — Depth-first: Inorder(Left-Root-Right), Preorder(Root-Left-Right), Postorder(Left-Right-Root). Inorder gives sorted sequence for BST. Preorder copies tree structure.
Big-O dominance rule — In O(n² + 5n + 100), only highest degree term matters: O(n²). Drop constants and lower terms for asymptotic analysis.