2022 amc 12b

Solution 2. Consider the 20 term sequence of 's and 's. Keeping all other terms 1, a sequence of consecutive 0's can be placed in locations. That is, there are 20 strings with 1 zero, 19 strings with 2 consecutive zeros, 18 strings with 3 consecutive zeros, ..., 1 string with 20 consecutive zeros. Hence there are strings with consecutive zeros.

2022 amc 12b. Solution 1. We can rewrite the given equation as . Hence, must be a power of and larger than . The first power of 2 that is larger than , namely , does satisfy the equation: . In fact, this is the only solution; is exponential whereas is linear, so their graphs will not intersect again. Now, let the common difference in the sequence be .

Resources Aops Wiki 2022 AMC 10B Problems/Problem 25 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2022 AMC 10B Problems/Problem 25. ... 2022 AMC 12B (Problems • Answer Key • Resources) Preceded by

The 2022 AMC 10A/12A will be held on Thursday, November 10, 2022. We posted the 2022 AMC 10A Problems and Answers, and 2022 AMC 12A Problems and Answers at 8:00 a.m. on November 11, 2022. Your attention would be very much appreciated. More details can be found at: Every Student Should Take Both the AMC 10A/12A and 10 B/12B!Notice that now Re and Im = Im. This forces and to be equal to and , meaning that all three are equally spaced along the unit circle. We can now show that we can construct complex numbers when that do not satisfy the conditions in the problem. Suppose that the condition in the problem holds for some . We can now add two points and anywhere on ... The test was held on Tuesday, November , . 2021 Fall AMC 12B Problems. 2021 Fall AMC 12B Answer Key. Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Resources Aops Wiki 2022 AMC 12B Problems/Problem 6 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2022 AMC 12B Problems/Problem 6. Redirect page. Redirect to: 2022 AMC 10B Problems/Problem 8;The first link contains the full set of test problems. The rest contain each individual problem and its solution. 2022 AMC 8 Problems. 2022 AMC 8 Answer Key. Problem 1. Problem 2. Problem 3. Problem 4. Problem 5.Resources Aops Wiki 2023 AMC 12B Problems/Problem 25 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2023 AMC 12B Problems/Problem 25. The following problem is from both the 2023 AMC 10B #25 and 2023 AMC 12B #25, so both problems redirect to this page.Download the 2022 AMC 10A Problems by clicking here or the 2022 AMC 12A Problems by clicking here! Learn how our curriculum can help from our expert student service advisors by phone at (949) 305-1705 or by email at [email protected]. Join 8 Week Prep Courses for final preparation for this fall's AMC 10+12!

2022 AMC 8 Results Just Announced — Ten Students Received Perfect Scores; It Is Easier to Qualify for the AIME Through the AMC 12 Than Through the AMC 10; 93 Students Qualified for the 2022 AIME and 2 Students Received Perfect Scores on the 2021 Fall AMC 10/12; Congratulations to Isabella Z. for Winning the Akamai AMC 12B AwardAnswer Key:1.A 2.D 3.A 4.B 5.B 6.B 7.D 8.D 9.B 10.D11.E 12.D 13.D 14.E 15.C16.C 17.D 18.C 19.A ...Resources Aops Wiki 2021 AMC 12B Problems/Problem 12 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2021 AMC 12B Problems/Problem 12. The following problem is from both the 2021 AMC 10B #19 and 2021 AMC 12B #12, so both problems redirect to this page.Solution Video to the following problems from the American Mathematics Competitions:2022 AMC 10B #62022 AMC 12B #3The following problem is from both the 2022 AMC 10B #8 and 2022 AMC 12B #6, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1 (Casework) 3 Solution 2 (Find A Pattern) 4 Solution 3 (Fastest) 5 Video Solution (🚀Under 3 min🚀) 6 Video Solution(1-16)Solution 3. We use a coordinates approach. Letting the origin be the center of the hexagon, we can let Then, and. We use the distance formula four times to get Thus, the perimeter of . ~sirswagger21. Note: the last part of this solution could have been simplified by noting that.

2002 AMC 12B problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. 2002 AMC 12B Problems. 2002 AMC 12B Answer Key. 2002 AMC 12B Problems/Problem 1. 2002 AMC 12B Problems/Problem 2. 2002 AMC 12B Problems/Problem 3.AMC System Announcement. MAA AMC launched a new competition administration platform, which hosts paper and digital formats of the competitions on a competition manager (CM) portal. CMs can conveniently administer the digital competition to students or print the paper competition with answer sheets and scan them to send them back.Hey guys, today we'll be covering the 2022 AMC 12B #25. Hope you enjoy!Solution Video to the following problems from the American Mathematics Competitions:2022 AMC 10B #252022 AMC 12B #23

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The 7th roots of unity strikes again. Pretty standard application of roots of unity.16 Nov 2023 ... Mastering AMC 10/12 book: https ... 2023 AIME Cutoff Predictions (and AMC 10A/12A, 10B/12B Difficulty Analysis) ... 2022). Rúbi Férez New 68K views.The first link contains the full set of test problems. The rest contain each individual problem and its solution. 2005 AMC 12B Problems. Answer Key. Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. First, measuring the radius of the circle obtains cm (when done on the paper version). Thus, any other measurement we get for the sides/diagonals should be divided by . Measuring the sides of the circle gets cm. The shorter diagonals are cm, and the longest diagonals measure cm. Thus, we'd like to estimate. Answers to the 2022 AMC 10B and 12B Exams held November 16, 2022 are available now. See all the answers and problem types from the exams! Continue reading. November 11, 2022 Contest Results. 2022 AMC 10A & AMC 12A Answer Key Released.

The following problem is from both the 2022 AMC 10B #2 and 2022 AMC 12B #2, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1; 3 Solution 2 (The Area Of A Triangle) 4 Video Solution 1; 5 Video Solution(1-16) 6 Video Solution by Interstigation; 7 See Also; Problem.The AMC 8 is administered from January 18, 2022 until January 24, 2022. According to the AMC policy, "problems and solutions are not discussed in any online or public forum until January 25," as emphasized in 2021-2022 AMC 8 Teacher's Manual. We posted the 2022 AMC 8 Problems and Answers at 11:59PM on Monday, January 24, 2022 Eastern Standard Time.Train for the AMC 12 with outstanding students from around the world in our AMC 12 Problem Series online class. LEARN MORE AMC 12 Problems and Solutions. AMC 12 problems and solutions. Year Test A Test B 2023: AMC 12A: AMC 12B: 2022: AMC 12A: AMC 12B: 2021 Fall: AMC 12A: AMC 12B: 2021 Spring: AMC 12A: AMC 12B: 2020: …The test was held on Wednesday, February 19, 2020. 2020 AMC 12B Problems. 2020 AMC 12B Answer Key. Problem 1. Problem 2. Problem 3. Problem 4.Solution 1 (Law of Cosines) Let . Since is the midpoint of , we must have . Since the centroid splits the median in a ratio, and . Applying Law of Cosines on and yields and . Finally, applying Law of Cosines on yields . The requested sum is .The test was held on Tuesday, November 14, 2023. 2023 AMC 12B Problems. 2023 AMC 12B Answer Key. Problem 1. Problem 2. Problem 3. Problem 4. Problem 5.AMC 12, at least the top 5% of all scorers on the AMC 12A and the top 5% of scorers on the AMC 12B are invited. The cutoff scores for AIME •qualification be announced will after each competition 10B, (10A,12A, and 12B) based onthe distributionofscores. There nois predetermined cutoff score for the 2022 AIME and this year's AIME cutoffResources Aops Wiki 2002 AMC 12B Problems Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. GET READY FOR THE AMC 12 WITH AoPS Learn with outstanding instructors and top-scoring students from around the world in our AMC 12 Problem Series online course.flag Report Content You should report content if: It may be offensive. There is something wrong with it (statement or difficulty value) It isn't original. Thanks for keeping the Math Contest Repository a clean and safe environment!Resources Aops Wiki 2012 AMC 12B Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2012 AMC 12B. 2012 AMC 12B problems and solutions. The test was held on February 22, 2012. The first link contains the full set of test problems. The rest contain each individual …

Solution 4 (Two Square Arrays) This solution refers to the Diagram section. As shown below, the taxicab distance between each red point and the origin is even, and the taxicab distance between each blue point and the origin is odd. Note that the red array consists of points, and the blue array consists of points. Together, the answer is.

Resources Aops Wiki 2022 AMC 12B Problems/Problem 12 Page. Article Discussion View source History. Toolbox. Recent ...AMC 12 B. Kentucky. Hannah E Laney. 11. The Gatton Academy Of Mathematics And Science. KY. AMC 12 B. Kentucky. Joanna T Cholewo.Get Mastering AMC 10/12 book: https://www.omegalearn.org/mastering-amc1012. The book includes video lectures for every chapter, formulas for every topic, and...From the MAA "IMPORTANT ANNOUNCEMENT REGARDING THE AMC 10/12 A. We are aware that elements of the AMC 10/12 A competition were leaked. We have been monitoring online channels and have already identified the source of the leak. This specific test site has been disqualified from this competition cycle and banned from participating in the MAA AMC ... Resources Aops Wiki 2022 AMC 12B Problems/Problem 12 Page. Article Discussion View source History. Toolbox. Recent ... Nov 18, 2022 · Solution Video to the following problems from the American Mathematics Competitions:2022 AMC 12B #25 The following problem is from both the 2022 AMC 10B #8 and 2022 AMC 12B #6, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1 (Casework) 3 Solution 2 (Find A Pattern) 4 Solution 3 (Fastest) 5 Video Solution (🚀Under 3 min🚀) 6 …The following problem is from both the 2022 AMC 10B #7 and 2022 AMC 12B #4, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1; 3 Solution 2; 4 Solution 3 (Pythagorean Triples) 5 Solution 4; 6 Video Solution (⚡️Lightning Fast⚡️) 7 Video Solution(1-16)

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The following problem is from both the 2022 AMC 10B #16 and 2022 AMC 12B #13, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1; 3 Solution 2 (Clever) 4 Solution 3 (Coordinate Geometry) 5 Solution 4; 6 Solution 5 (Fastest Similar Triangles) 7 Video Solution by mop 2024;Comment below if you have any questions, related to this one or not!More 2022 AMC 10B Solutions: https://bit.ly/2022AMC10B2021 Fall AMC 10B Solutions: https:...Answers to the 2022 AMC 10B and 12B Exams held November 16, 2022 are available now. See all the answers and problem types from the exams! Continue reading. November 11, 2022 Contest Results. 2022 AMC 10A & AMC 12A Answer Key Released.2022 AMC 12B (Problems • Answer Key • Resources) Preceded by Problem 8: Followed by Problem 10: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 …2022 AMC 12B Problem #24Solution 1. Draw diagonals and to split the pentagon into three parts. We can compute the area for each triangle and sum them up at the end. For triangles and , they each have area . For triangle , we can see that and . Using Pythagorean Theorem, the altitude for this triangle is , so the area is . Adding each part up, we get .The test was held on February 13, 2019. 2019 AMC 12B Problems. 2019 AMC 12B Answer Key. Problem 1. Problem 2. Problem 3. Problem 4.The following problem is from both the 2022 AMC 10B #8 and 2022 AMC 12B #6, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1 (Casework) 3 Solution 2 (Find A Pattern) 4 Solution 3 (Fastest) 5 Video Solution (🚀Under 3 min🚀) 6 …AoPS Community 2022 AMC 12/AHSME 4 The least common multiple of a positive integer nand 18 is 180, and the greatest common divisor of nand 45 is 15. What is the sum of the digits of n? (A) 3 (B) 6 (C) 8 (D) 9 (E) 12 5 Let the taxicab distance between points (x 1,y 1) and (x 2,y 2) in the coordinate plane is given by |x 1 −x 2|+ |y 1 −y 2|.For how many …About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright ... ….

The AMC-10/12A Contest will be held at FAU on Wednesday, November 8, 2023, beginning at 3:00pm. The AMC-10/12B Contest will be held at FAU on Tuesday, November 14, 2023, beginning at 8:00am. Please note that while only in-person proctoring by FAU is permitted, contests will be given via the computerized format at the indicated time.2022 AMC 12B Problems/Problem 15. Redirect page. 2022 AMC 10B Problems/Problem 17. Art of Problem Solving is an. ACS WASC Accredited School.Dec 19, 2023 - Jan 11, 2024. $113.00. Final day to order additional bundles for the 8. Jan 11, 2024. AMC 8 Competition: Jan 18 - 24, 2024.AIME 자료 공유에 힘입어. 2023년 새해를 맞아 AMC12 시험을 준비하는 학생들을 위해. 2000-2022년도 모든 기출문제를 정리한 PDF파일을 공유합니다. . . . *AIME 자료가 필요하시다면 아래 링크의 이전 게시글을 확인해주세요. AIME 2000~2022 기출문제 PDF 자료 공유. 안녕하세요.Solution Video to the following problems from the American Mathematics Competitions:2022 AMC 10B #82022 AMC 12B #6Resources Aops Wiki 2022 AMC 12B Problems/Problem 13 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2022 AMC 12B Problems/Problem 13. Redirect page. Redirect to: 2022 AMC 10B Problems/Problem 16;Answers to the 2022 AMC 10B and 12B Exams held November 16, 2022 are available now. See all the answers and problem types from the exams! Continue reading. November 11, 2022 Contest Results. 2022 AMC 10A & AMC 12A Answer Key Released.Resources Aops Wiki 2019 AMC 12B Problems Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. TEXTBOOKS FOR THE AMC 12 For over 25 years, students have used Art of Problem Solving textbooks as a central part of their AMC preparation. ...Exactly the week before exam of AMC 10A and 12A 2023 I released 4 preparation videos(links below) that had useful ideas for AMC 10 12 and other exams and I s...The following problem is from both the 2022 AMC 10B #23 and 2022 AMC 12B #22, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1 (Geometric Probability) 3 Solution 2 (Generalization and Induction) 4 Solution 3 (Observations) 5 Remark (Calculus) 2022 amc 12b, Solution 2 (Complex Numbers) We write as We'd like to rotate about which is in the complex plane, by an angle of counterclockwise. The formula for rotating the complex number about the complex number by an angle of counterclockwise is given as Plugging in our values , we evaluate the expression as which corresponds to on the Cartesian plane., 2022 AMC 12B Problems/Problem 15. Redirect page. 2022 AMC 10B Problems/Problem 17. Art of Problem Solving is an. ACS WASC Accredited School., The 2021 AMC 10B/12B (Fall Contest) will be held on Tuesday, November 16, 2021. We posted the 2021 AMC 10A (Fall Contest) Problems and Answers, and 2021 AMC 12A (Fall Contest) Problems and Answers at 8:00 a.m. on November 17, 2021 . Your attention would be very much appreciated. More details can be found at: Every Student Should Take Both the ..., Resources Aops Wiki 2022 AMC 12B Problems/Problem 5 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2022 AMC 12B Problems/Problem 5. Contents. 1 Problem; 2 Solution 1 (Cartesian Plane) 3 Solution 2 (Complex Numbers) 4 Video Solution 1;, AMC 12 Click “ here ” to download 2023 AMC 12A problems and answer key. Click “ here ” to download 2023 AMC 12B problems and answer key. Click “ here ” to download 2022 AMC 12A problems and answer key. Click “ here ” to download 2022 AMC 12B problems and answer key., 2020 AMC 12A Problems Problem 1 Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left? Problem 2 The acronym AMC is shown in the rectangular grid below with grid lines spaced unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC Problem 3, 2022 AMC 10B Problems/Problem 1. The following problem is from both the 2022 AMC 10B #1 and 2022 AMC 12B #1, so both problems redirect to this page., Solution 4. It is obvious to put , , and on the faces that share the same vertex. As is the next biggest number, the face with has to be next to the faces with and . As is the next biggest number, the face with has to be next to the faces with and . making the face with next to the faces with and . Therefore the answer is., The following problem is from both the 2018 AMC 12B #10 and 2018 AMC 10B #14, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1 (Statistics) 3 Solution 2 (Algebra) 4 Video Solution (HOW TO THINK CRITICALLY!!!) 5 See Also; Problem. A list of positive integers has a unique mode, which occurs exactly times. What is the ..., Solution 2. Let be the bottom base, be the top base, be the height of the bottom triangle, be the height of the top triangle. Thus, so Let so we get This gives us a quadratic in ie. so. - Solution by MathAwesome123, added by ccx09., Resources Aops Wiki 2022 AMC 12B Problems/Problem 16 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2022 AMC 12B Problems/Problem 16. Contents. 1 Problem; 2 Solution; 3 Solution 2; 4 Solution 3; 5 Video Solution by mop 2024; 6 Video Solution (Just 2 min!), Comment below if you have any questions, related to this one or not!More 2022 AMC 10B Solutions: https://bit.ly/2022AMC10B2021 Fall AMC 10B Solutions: https:..., The first link contains the full set of test problems. The rest contain each individual problem and its solution. 2004 AMC 12B Problems. Answer Key. 2004 AMC 12B Problems/Problem 1. 2004 AMC 12B Problems/Problem 2. 2004 AMC 12B Problems/Problem 3. 2004 AMC 12B Problems/Problem 4. 2004 AMC 12B Problems/Problem 5., Resources Aops Wiki 2021 AMC 12B Problems Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. GET READY FOR THE AMC 12 WITH AoPS Learn with outstanding instructors and top-scoring students from around the world in our AMC 12 Problem Series online course. ..., 17 Oct 2023 ... AMC 2023 2024 12A 12B 10A 10B Solutions Problems American Mathematics Competitions Tutor Preparation. Math Gold Medalist•1.8K views · 9:59 · Go ...., GET READY FOR THE AMC 10! Join top-scoring peers and outstanding instructors in our online AMC 10 Problem Series course. CHECK SCHEDULE 2021 Fall AMC 10B Problems. 2021 Fall AMC 10B Printable versions: Wiki • Fall AoPS Resources • Fall PDF: Instructions. This is a 25-question, multiple choice test. ... 2022 AMC 10A: 1 ..., 202 1 AMC 12 A Problems Problem 1 What is the value of t 5 > 6 > 7 F :t 5 Et 6 Et 7 ;ë Problem 2 Under what conditions is ¾ = 6 E> 6 L = E> true, where = and > are real numbers? (A) It is never true. (B) It is true if and only if => L rä (C) It is true if and only if = E> R rä (D) It is true if and only if => L r and = E> R rä, The best film titles for charades are easy act out and easy for others to recognize. There are a number of resources available to find movie titles for charades including the AMC F..., 8 Sept 2022 ... However, you can do what I'm doing and take AMC 10A and then the AMC 12B. The qualification is different for each test(for example, it could be ..., 2022 AM 12. 2022 AM 12. The problems in the AM-Series ontests are copyrighted by American Mathematics ompetitions at Mathematical Association of America (www.maa.org). For more practice and resources, visit ziml.areteem.org. Q u e s t i o n., The test was held on February 17, 2016. 2016 AMC 12B Problems. 2016 AMC 12B Answer Key. Problem 1. Problem 2. Problem 3. Problem 4., The AMC 10 and AMC 12 Have 10-15 Questions in Common. All students should take both the A-date and B-date AMC tests. The AMC 10B/12B gives a student a second chance to qualify for the American Invitational Mathematics Exam ( AIME ). If a student does not qualify for the AIME through the AMC10A/12A, then he/she can qualify …, Registration for the AIME is automatic. Any students taking the AMC 12 and scoring in the top 5% or over 100, or are in the top 2.5% of the scores on the AMC 10 qualify. The testing materials (including the tests, answer sheets, teachers manual, and computer identification form) are included with the results packet from the AMC 10 and/or the ..., CHECK SCHEDULE 2021 AMC 12B Problems Contents 9/26/22, 4:03 PM Art of Problem Solving 2/7 26 See also What is the ratio of the sum of the odd divisors of to the sum of the even divisors of How many integer values of satisfy Solution At a math contest, students are wearing blue shirts, and another students are wearing yellow shirts., Resources Aops Wiki 2022 AMC 12B Problems/Problem 7 Page. Article Discussion View source History. Toolbox. Recent changes Random page Help What links here Special pages. Search. 2022 AMC 12B Problems/Problem 7. Redirect page. Redirect to: 2022 AMC 10B Problems/Problem 10;, Solution 2 (unnecessary numerical values) Given that the first three glasses are full and the fourth is only full, let's represent their contents with a common denominator, which we'll set as 6. This makes the first three glasses full, and the fourth glass full. To equalize the amounts, Mrs. Jones needs to pour juice from the first three ..., 收藏. 1. 知识. 校园学习. 美国留学. amc. 陈教授教育学院. 2022 AMC 10A第25题. 2022 AMC 12A第25题, 视频播放量 61、弹幕量 0、点赞数 1、投硬币枚数 0、收藏人数 0、转发人数 1, 视频作者 陈教授教育学院, 作者简介 ,相关视频:2022 AMC 10A第25题,2022 AMC 10B 第25题,12B第23题 ..., The following problem is from both the 2021 AMC 10B #25 and 2021 AMC 12B #25, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1; 3 Solution 2. 3.1 Diagram; 4 Solution 3; 5 Solution 4; 6 Solution 5; 7 Remark; 8 Video Solution; 9 Video Solution by Interstigation (In-Depth, Straight-forward), Here are the overall results for the 2022 AMC 10A, AMC 12A, AMC 10B, and AMC 12B contests at Bard College: School AMC 12A Statistics. Average score for entire school is: 94.8. Average score for grade 11 is: 100.0 (3 Students) Average score for grade 10 is: 91.8 (5 Students) Average score for grade 9 is: 94.5 (2 Students), flag Report Content You should report content if: It may be offensive. There is something wrong with it (statement or difficulty value) It isn't original. Thanks for keeping the Math Contest Repository a clean and safe environment!, The 2023 AMC 10A/12A will be held on Wednesday, November 8, 2023. We posted the 2023 AMC 10A Problems and Answers, and 2023 AMC 12A Problems and Answers at 8:00 a.m. (EST) on November 9, 2023. Your attention would be very much appreciated. More details can be found at: Every Student Should Take Both the AMC 10A/12A and 10 B/12B!, Get Set for the 2022-23 American Mathematics Competition 10/12 Y2023. ... Hence candidates CANNOT register for both AMC 10A and 12A but they can register for AMC 10A and 12B. The AMC 10/12 is a 75-minute 25 MCQ question competition that seeks to give students an exposure to mathematics that is ‘novel’ and ‘out-of-the-box’. AMC …, The following problem is from both the 2022 AMC 10B #17 and 2022 AMC 12B #15, so both problems redirect to this page. Contents. 1 Problem; 2 Solution 1 (Modular Arithmetic) 3 Solution 2 (Factoring) 4 Solution 3 (Elimination) 5 Solution 3a (Elimination) 6 Solution 3b (Elimination + Number Theory)