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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 ## If a is an integer, what is the units digit of a^18? (1) a^ ##### This topic has 2 expert replies and 0 member replies ### Top Member ## If a is an integer, what is the units digit of a^18? (1) a^ ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult If a is an integer, what is the units digit of a^18? (1) a^2 has a units digit of 9 (2) a^7 has a units digit of 3 OA D Source: Veritas Prep ### 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 04 Dec 2012 Posted: 2033 messages Followed by: 238 members Upvotes: 1443 BTGmoderatorDC wrote: If a is an integer, what is the units digit of a^18? (1) a^2 has a units digit of 9 (2) a^7 has a units digit of 3 OA D Source: Veritas Prep If we want to know the units digit of a^18, we must know something about the units digit of a. (1) a^2 has a units digit of 9 Test digits to see if there is more than one units digit that will yield a 9 when squared: a = 1 --> a² = 1 --> no. a = 2 --> a² = 4 --> no. a = 3 --> a² = 9 --> YES. a = 4 --> a² = 16 --> no. a = 5 --> a² = 25 --> no. a = 6 --> a² = 36 --> no. a = 7 --> a² = 49 --> YES a = 8 --> a² = 64 --> no. a = 9 --> a² = 81 --> no. Thus, 'a' can have a units digit of 3 or 7. For each of these, though, a^18 = (a^2)^9 . So we'd be taking a number with a units digit of 9 and taking it to the 9th power. This will always yield a units digit of 9. (Odd powers of 9 always end in a 9). Sufficient. (2) a^7 has a units digit of 3 Units digit patterns always repeat after every 4th power. Eg: x = 2 x^2 = 4 x^3 = 8 x^4 = 16 x^5 = 32 x^6 = 64 etc. Some digits (0, 1, 5, and 6) always stay constant, and some (4 and 9) repeat every 2 powers, but every digit repeats at least every 4th power. So, every a^(4n + 3) will have the same units digit. Or more simply put, a^7 will have the same units digit as a^3. Let's just test a^3 for every digit (ignoring everything except units digits): a = 1 --> a^3 = 1 --> no. a = 2 --> a^3 = 8 --> no. a = 3 --> a^3 = 7 --> no. a = 4 --> a^3 = 4 --> no. a = 5 --> a^3 = 5 --> no. a = 6 --> a^3 = 6 --> no. a = 7 --> a^3 = 3 --> YES. a = 8 --> a^3 = 2 --> no a = 9 --> a^3 = 9 --> no. The units digit of 'a' must be 7. This is sufficient to determine the units digit of a^18. The answer is D. _________________ 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. • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 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 • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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