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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 ## 17^27 has a units digit of: ##### This topic has 4 expert replies and 0 member replies ### Top Member ## 17^27 has a units digit of: ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult 17^27 has a units digit of: (a) 1 (b) 2 (c) 3 (d) 7 (e) 9 OA C Source: Manhattan Prep ### GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2424 messages Followed by: 18 members Upvotes: 43 Top Reply BTGmoderatorDC wrote: 17^27 has a units digit of: (a) 1 (b) 2 (c) 3 (d) 7 (e) 9 Since we only care about units digits, we can rewrite the expression as: 7^27 Now we can evaluate the pattern of the units digits of 7^n for positive integer values of n. That is, letâ€™s look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are ONLY concerned with the units digit of 7 raised to each power. 7^1 = 7 7^2 = 9 7^3 = 3 7^4 = 1 7^5 = 7 The pattern of the units digit of powers of 7 repeats every 4 exponents. The pattern is 7-9-3-1. In this pattern, all positive exponents that are multiples of 4 will produce a 1 as its units digit. Thus: 7^28 has a units digit of 1, and 7^27 has a units digit of 3. Answer: C _________________ Scott Woodbury-Stewart Founder and CEO scott@targettestprep.com See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 Top Reply BTGmoderatorDC wrote: 17^27 has a units digit of: (a) 1 (b) 2 (c) 3 (d) 7 (e) 9 Source: Manhattan Prep $$? = \left\langle {{{17}^{27}}} \right\rangle = \left\langle {{7^{27}}} \right\rangle$$ $$\left\langle {{7^4}} \right\rangle = \left\langle {{7^2} \cdot {7^2}} \right\rangle = \left\langle {\left\langle {{7^2}} \right\rangle \cdot \left\langle {{7^2}} \right\rangle } \right\rangle = 1$$ $$\left\langle {{7^{24}}} \right\rangle = \left\langle {{7^4} \cdot {7^4} \cdot \ldots \cdot {7^4}} \right\rangle = {\left\langle {{7^4}} \right\rangle ^6} = 1$$ $$? = \left\langle {{7^{24}} \cdot {7^3}} \right\rangle = \left\langle {{7^{24}}} \right\rangle \cdot \left\langle {{7^3}} \right\rangle = 1 \cdot 3 = 3$$ 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 ### GMAT/MBA Expert GMAT Instructor Joined 22 Aug 2016 Posted: 1890 messages Followed by: 30 members Upvotes: 470 BTGmoderatorDC wrote: 17^27 has a units digit of: (a) 1 (b) 2 (c) 3 (d) 7 (e) 9 OA C Source: Manhattan Prep Let's understand the power cycle of 7. â€¢ 7^1 = 7; unit digit = 7; â€¢ 7^2 = 49; unit digit = 9; â€¢ 7^3 = 343; unit digit = 3; â€¢ 7^4 = ...1; unit digit = 1; â€¢ 7^5 = ...7; unit digit = 7; We see that units digit of the power of 7 repeats after every 4 cycle. Let's see 17^27 now. 17^27 = 17^(24 + 3) = 17^(6*4 + 3) Note that units digit of 17^27 will be the same that of 17^3 since 6*4 has a cycle of 4. Again, the units digit of 17^3 will have the same units digit as has 7^3. Now 7^3 = 343; thus, the units digit of 17^27 = 3. The correct answer: C Hope this helps! -Jay _________________ Manhattan Review Locations: Manhattan Review Mumbai | GMAT Prep New Delhi | GRE Prep Malleswaram | Vijayawada GRE Coaching | and many more... Schedule your free consultation with an experienced GMAT Prep Advisor! Click here. ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12827 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 BTGmoderatorDC wrote: 17^27 has a units digit of: (a) 1 (b) 2 (c) 3 (d) 7 (e) 9 Look for a pattern 17^1 = 17 17^2 = (17)(17) = ---9 [aside: we need not determine the other digits. All we care about is the units digit] 17^3 = (17)(17^2) = (17)(---9) = ----3 17^4 = (17)(17^3) = (17)(---3) = ----1 17^5 = (17)(17^4) = (17)(---1) = ----7 NOTICE that we're back to where we started. 17^5 has units digit 7, and 17^1 has units digit 7 So, at this point, our pattern of units digits keep repeating 7, 9, 3, 1, 7, 9, 3, 1, . . . We say that we have a "cycle" of 4, which means the digits repeat every 4 powers. So, we get: 17^1 = --7 17^2 = ---9 17^3 = ----3 17^4 = ----1 17^5 = ----7 17^6 = ---9 17^7 = ----3 17^8 = ----1 17^9 = ----7 17^10 = ----9 etc. Notice that when the exponent is a MULTIPLE of 4 (4, 8, 12, 16, ...), the units digit will be 1 Since 24 is a MULTIPLE of 4, we know that the units digit of 17^24 will be 1 Continuing on, we get: 17^25 = ----7 17^26 = ---9 17^27 = ----3 Answer: C Here's an article I wrote on this topic (with additional practice questions): https://www.gmatprepnow.com/articles/units-digits-big-powers Cheers, Brent _________________ Brent Hanneson â€“ 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 GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! • Award-winning private GMAT tutoring Register now and save up to$200

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