1582 articles
Eulers totient function phi(n) counts integers up to n coprime to n and underpins RSA, modular inverse for composite moduli, and Eulers theorem. Master the prime-factorization formula, the sieve variant in O(N log log N), and the multiplicative-function tricks that make phi indispensable for competitive programming.
Computing C(n, r) modulo a prime appears in nearly every counting problem at Google and Meta. Master Pascal triangle for small n, precomputed factorials with modular inverse for n up to 10^6, and Lucas theorem for n up to 10^18.
Number theory basics — divisibility, congruences, digit DP, perfect numbers — power half of the math problems at Amazon and Microsoft. Master the divisibility rules, modular congruence properties, and digit-extraction tricks before tackling advanced number theory.
Matrix exponentiation collapses any linear recurrence into O(log n) by raising a transition matrix to the n-th power. Master Fibonacci, k-th order recurrences, and DP optimization tricks that let you answer queries with n up to 10^18 in milliseconds.
Bit manipulation is the cheat code of competitive programming and tier-1 interviews. Master the XOR identities, n & (n-1) tricks, subset enumeration over a bitmask, and bitmask DP techniques that turn O(2^n) brute force into elegant constant-factor wins.