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- ===Arithmetic=== {{main|Arithmetic}}6 KB (902 words) - 17:16, 22 October 2024
- ...vided: an elementary one that rests close to basic principles of [[modular arithmetic]], and an elegant method that relies on more powerful [[algebra]]ic tools. == Problems ==4 KB (639 words) - 00:53, 2 February 2023
- ...ath>b</math>, <cmath>n | (a - b).</cmath> We may rewrite this in [[modular arithmetic]] as <math> a - b \equiv 0 \textrm{ } (\textrm{mod }n)</math>, or <cmath>a == Olympiad Problems ==11 KB (1,986 words) - 18:13, 19 June 2024
- *[[Modular arithmetic]] ...visor|divisors]], and more). Also includes [[base number]]s and [[modular arithmetic]].3 KB (399 words) - 22:08, 8 January 2024
- ...special type of arithmetic that involves only [[integers]]. Since modular arithmetic is such a broadly useful tool in [[number theory]], we divide its explanati * [[Modular arithmetic/Introduction|Introduction to modular arithmetic]]992 bytes (121 words) - 12:25, 20 December 2024
- ...[number theory]] for simplifying the computation of exponents in [[modular arithmetic]] (which students should study more at the introductory level if they have In contest problems, Fermat's Little Theorem is often used in conjunction with the [[Chinese Re16 KB (2,660 words) - 22:42, 28 August 2024
- ==Problems== *([[2018 AMC 10B Problems/Problem 16|2018 AMC 10B]]) Let <math>a_1,a_2,\dots,a_{2018}</math> be a str3 KB (542 words) - 16:45, 21 March 2023
- == Problems == * Practice Problems on [https://artofproblemsolving.com/alcumus/ Alcumus]10 KB (1,572 words) - 21:11, 22 September 2024
- ===Modular Arithmetic=== Modular arithmetic can help determine if a number is not prime.6 KB (1,036 words) - 17:26, 2 September 2024
- == Example Problems == *[[Modular arithmetic]]4 KB (547 words) - 16:23, 30 December 2020
- ...ed value is the discrete logarithm, used in [[cryptography]] via [[modular arithmetic]]. It's the lowest value <math>c</math> such that <math>a^c=mx+b</math> for == Problems ==4 KB (680 words) - 11:54, 16 October 2023
- ...lution or solutions to a Diophantine equation is closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely === Modular Arithmetic ===9 KB (1,434 words) - 00:15, 4 July 2024
- Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and in all of mathematics. ...ons to the equation <math>n = q_1 + \ldots + q_c \pmod p</math> in modular arithmetic, subject to the constraints<math>q_1,\ldots,q_c \neq 0 \pmod p</math>. Thi7 KB (1,184 words) - 18:44, 7 December 2024
- ...fficult and more interesting problems that are easily solved using modular arithmetic. ==Understand Modular Arithmetic==16 KB (2,406 words) - 07:56, 10 July 2024
- ...quations]], testing whether certain large numbers are prime, and even some problems in cryptology. == Arithmetic Modulo n ==14 KB (2,317 words) - 18:01, 29 October 2021
- Now, consider the powers of <math>2</math> [[modular arithmetic | mod]] <math>9</math>: <math>2^{6n} \equiv 1, 2^{6n + 1} \equiv 2, 2^{6n + [[Category:Intermediate Number Theory Problems]]8 KB (1,283 words) - 18:19, 8 May 2024
- ...terms <math> a_{2n}, a_{2n+1}, </math> and <math> a_{2n+2} </math> are in arithmetic progression. Let <math> a_n </math> be the greatest term in this sequence t ...e find that by either the [[quadratic formula]] or trial-and-error/modular arithmetic that <math>x=5</math>. Thus <math>f(n) = 4n+1</math>, and we need to find t3 KB (547 words) - 10:41, 7 August 2024
- ==Solution 5 (Easy Modular Arithmetic)== [[Category:Intermediate Number Theory Problems]]2 KB (338 words) - 18:56, 15 October 2023
- These are the solutions to the problems related to the '''[[Pigeonhole Principle]]'''. ...e difference to be a multiple of 7, the integers must have equal [[Modular arithmetic/Introduction |modulo]] 7 residues. To avoid having 15 with the same residue10 KB (1,617 words) - 00:34, 26 October 2021
- If the [[modular arithmetic|remainder]] of the division of <math>a</math> with <math>35</math> is <math [[Category:Introductory Algebra Problems]]628 bytes (84 words) - 15:22, 6 May 2007