← Topic 2 · Number Theory
AMC 10/12 · Cheatsheet

Topic 2 · Number Theory

Chapter 1 · Modular arithmetic & divisors

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Congruence
a≡b(modm)  ⟺  m∣(a−b)a \equiv b \pmod m \iff m \mid (a-b)
Operations
Add, subtract, multiply congruences freely; division only by residues coprime to mm.
Last digit
Powers cycle mod 10; reduce the exponent by the cycle length.
Divisor count
n=∏piai⇒d(n)=∏(ai+1)n=\prod p_i^{a_i} \Rightarrow d(n)=\prod (a_i+1)
Perfect squares
Odd number of divisors ⟺ perfect square.