Modular arithmetic

12 curated · 6 from open sources · 196.6 million generated

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Two integers are congruent modulo $n$ , written $a mod n$ , when they leave the same remainder upon division by $n$ . Congruences behave like equations: you may add, subtract, and multiply both sides by the same integer, and replace any number by its remainder modulo $n$ at any time. For a prime $p$ , Fermat's little theorem states $a^{p-1} mod p = 1$ whenever $p$ does not divide $a$ . To solve $ax mod n = b$ , multiply both sides by the inverse of $a$ modulo $n$ , the number $a^{- 1}$ satisfying $a imes a^{-1} mod n = 1$ .

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Powers modulo nMedium · 196.6 million variations

Curated

#	Title	Difficulty	Source	Tags
579	A big power mod fiveModular arithmetic	Easy	Lemma	Modular arithmetic
580	A linear congruenceModular arithmetic	Easy	Lemma	Modular arithmetic
581	Two remainders at onceModular arithmetic	Easy	Lemma	Modular arithmetic
582	Last digit of a power of threeModular arithmetic	Easy	Lemma	Modular arithmetic
585	Squares mod sevenModular arithmetic	Easy	Lemma	Modular arithmetic
61	One more than a common multipleRemainders	Medium	Lemma	Least common multipleRemainders
583	Fermat's little theoremModular arithmetic	Medium	Lemma	Modular arithmetic
584	Wilson's theoremModular arithmetic	Medium	Classical	Modular arithmetic
1368	Remainders with 39 and 56Modular arithmetic	Medium	Hall & Knight	Modular arithmeticChinese remainder theorem
1785	General form of remaindersModular arithmetic	Medium	Hall & Knight	Chinese remainder theoremModular arithmetic
1786	Smallest integers with given remaindersModular arithmetic	Medium	Hall & Knight	Chinese remainder theoremModular arithmetic
59	A giant power, tamedModular arithmetic	Deep	Lemma	Fermat's little theorem

From open sources

Showing everything, including problems with no answer to check, a missing figure, or a near-copy elsewhere. Show only complete problems

#	Title	Difficulty
128015	Clock Arithmetic · Example 1	Easy
128016	Clock Arithmetic · Example 2	Easy
128018	Clock Arithmetic · Example 4	Easy
128019	Clock Arithmetic · Example 5	Easy
128020	Clock Arithmetic · Example 6	Easy
128017	Clock Arithmetic · Example 3	Easy