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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## What is the remainder if 7^10 is divided by 100? ##### This topic has 6 expert replies and 1 member reply ### Top Member ## What is the remainder if 7^10 is divided by 100? ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 OA E Source: Manhattan Prep ### GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2244 messages Followed by: 17 members Upvotes: 43 BTGmoderatorDC wrote: What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 The remainder when 7^10 is divided by 100 is equal to the last two digits of the expansion of 7^10. Notice that 7^2 = 49, or 50 - 1. So 7^4 = (50 - 1)^2 = 2500 - 100 + 1 = 2401. The last two digits of 7^2 is 49 and those of 7^4 is 01 or simply 1. Since 7^10 = 7^2 x 7^4 x 7^4, the last two digits of 7^10 is 49 x 1 x 1 = 49 Answer: E ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1440 messages Followed by: 32 members Upvotes: 59 BTGmoderatorDC wrote: What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 Source: Manhattan Prep First note that: > 1 is the remainder of 101 (=1*100+1) divided by 100 > 32 is the remainder of 532 (=5*100+32) divided by 100 > 47 is the remainder of 7847 (=78*100+47) divided by 100 $${7^{10}} = K \cdot 100 + R{\mkern 1mu} {\mkern 1mu} \,\,{\mkern 1mu} {\mkern 1mu} \left( {K\,\,{\mathop{\rm int}} \,\,,\,\,\,0 \le R \le 99\,\,{\mathop{\rm int}} } \right){\mkern 1mu}$$ $$? = R$$ $${7^{10}} = {\left( {{7^2}} \right)^5} = {49^5}$$ $${49^2} = {\left( {50 - 1} \right)^2} = {5^2} \cdot {10^2} - 100 + 1 = M \cdot 100 + 1\,\,\,,\,\,\,M\,\,{\mathop{\rm int}} \ge 1\,\,\,\,\,\,\,\,\,\,\left( {M = {5^2} - 1} \right)$$ $${49^4} = {\left( {M \cdot 100 + 1} \right)^2} = {M^2} \cdot {10^4} + M \cdot 200 + 1 = N \cdot 100 + 1\,\,\,,\,\,\,\,N\,\,{\mathop{\rm int}} \,\, \ge 1\,\,\,\,\,\,\,\,\left( {N = {M^2} \cdot {{10}^2} + 2M} \right)$$ $${49^5} = \left( {N \cdot 100 + 1} \right) \cdot 49 = K \cdot 100 + 49\,\,\,,\,\,\,\,K\,\,{\mathop{\rm int}} \,\, \ge 1\,\,\,\left( {K = 49N} \right)$$ $$? = 49$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br Last edited by fskilnik@GMATH on Sun Oct 14, 2018 12:03 pm; edited 1 time in total Junior | Next Rank: 30 Posts Joined 03 Oct 2018 Posted: 10 messages If we are asked to calculate the remainder when 6*5 is divided by 4, we observe that we can divide each of the numbers 6 and 5 and multiply their remainders to get the required answer. For example , $$\frac{6}{4}$$ yields a remainder of 2 whereas $$\frac{5}{4}$$ gives 1. Multiplying 2 and 1 is 2 - the same as the remainder when 30 is divided by 4. We use this same principle in the above problem. We break the given product into small numbers whose remainder we can easily find. In this case we can write it as $$7^9\cdot7$$ = $$343^3\cdot7$$ When 343 is divided by 100, the remainder is 43. Our required answer is (43*43)*(43*7) =1849 * 301 The remainders when 1849 and 301 are divided by 100 are 49 and 1. The product of 49 and 1 is 49 ... which is the required answer since 49 divided by 100 would continue to yield a remainder of 49. ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15203 messages Followed by: 1861 members Upvotes: 13060 GMAT Score: 790 BTGmoderatorDC wrote: What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 When a positive integer is divided by 100, the remainder is yielded by the last two digits: 123/100 = 1 R23 548/100 = 5 R48 692/100 = 6 R92 Thus: The remainder when 7¹⁰ is divided by 100 is equal to the last two digits of 7¹⁰. Calculate the last two digits for consecutive powers of 7 and look for a pattern: 7¹ --> 07 7² --> 49 7³ --> 43 7⁴ --> 01 7⁵ --> 07 The last two digits appear in a CYCLE OF 4: 07, 49, 43, 01...07, 49, 43, 01... Implication: When 7 is raised to a power that is a MULTIPLE OF 4 -- constituting the end of a cycle -- the last two digits will be 01. From there, the cycle will repeat: 07, 49, 43, 01... Since 8 is a multiple of 4, the last two digits for 7⁸ are 01. The cycle then repeats: 7⁹ ---> 07 7¹⁰ --> 49 The correct answer is E. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. ### GMAT/MBA Expert GMAT Instructor Joined 04 Dec 2012 Posted: 2033 messages Followed by: 238 members Upvotes: 1443 _________________ Ceilidh Erickson Manhattan Prep GMAT & GRE instructor EdM in Mind, Brain, and Education Harvard Graduate School of Education Manhattan Prep instructors all have 99th+ percentile scores and expert teaching experience. Sign up for a FREE TRIAL, and learn why we have the highest ratings in the GMAT industry! Free Manhattan Prep online events - The first class of every online Manhattan Prep course is free. Classes start every week. ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12744 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 Sweeeeeeeeeeeeet solution, Scott!! Cheers, Brent _________________ Brent Hanneson – Creator of GMATPrepNow.com Use our video course along with Sign up for our free Question of the Day emails And check out all of our free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months! ### GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2244 messages Followed by: 17 members Upvotes: 43 Brent@GMATPrepNow wrote: Sweeeeeeeeeeeeet solution, Scott!! Cheers, Brent Thanks Brent! • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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