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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 Find the units digit of tagged by: swerve This topic has 1 expert reply and 0 member replies Top Member Find the units digit of Timer 00:00 Your Answer A B C D E Global Stats Difficult $$\text{Fin the units digit of } 556^{17n}+339^{5m+15n},\text{ where }m\text{ and }n \text{ are positive integers.}$$ $$\text{1) }4m + 12m = 360.$$ $$\text{2) } n \text{ is the smallest 2-digit positive integer divisible by }5.$$ The OA is A Source: e-GMAT GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 swerve wrote: Find the units digit of 556^{17n}+339^{5m+15n} , where m and n are positive integers. (1) 14m + 12n = 360. (2) n is the smallest 2-digit positive integer divisible by 5. Source: e-GMAT $$\left\langle M \right\rangle \,\, = \,\,{\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,M$$ $$\left\langle {{{556}^n}} \right\rangle = \left\langle {{6^n}} \right\rangle = 6\,\,,\,\,\forall n \ge 1\,\,{\mathop{\rm int}}$$ $$\left\langle {{{339}^k}} \right\rangle = \left\langle {{9^k}} \right\rangle = \left\{ \matrix{ \,\,9\,\,,\,\,\forall k \ge 1\,\,{\rm{odd}} \hfill \cr \,\,1\,\,,\,\,\forall k \ge 2\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{{339}^{5\left( {m + 3n} \right)}}} \right\rangle = \left\{ \matrix{ \,\,9\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,5\,\,{\rm{odd}} \hfill \cr \,\,1\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,4\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\left( {m,n\,\, \ge 1\,\,{\rm{ints}}} \right)$$ $$\left\langle {{{556}^{17n}} + {{339}^{5\left( {m + 3n} \right)}}} \right\rangle \,\,\, = \,\,\,\left\{ \matrix{ \,\,\left\langle {6 + 9} \right\rangle = 5\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,5\,\,{\rm{odd}} \hfill \cr \,\,\left\langle {6 + 1} \right\rangle = 7\,\,\,,\,\,\forall \,m + 3n\,\, \ge \,\,4\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\left( {m,n\,\, \ge 1\,\,{\rm{ints}}} \right)$$ $$?\,\, = \left\langle {{{556}^{17n}} + {{339}^{5\left( {m + 3n} \right)}}} \right\rangle \,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,?\,\,\,:\,\,\,m + 3n\,\,\,{\rm{odd/even}}\,\,\,\,\,\,\,\,\,\,\left[ {\,m,n\,\, \ge 1\,\,{\rm{ints}}\,} \right]\,$$ $$\left( 1 \right)\,\,\,4m + 12n = 360\,\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 90\,\,\left( {{\rm{even}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,\left( {? = 7} \right)$$ $$\left( 2 \right)\,\,n = 10\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,m = 1\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 31\,\,\left( {{\rm{odd}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 5\,\, \hfill \cr \,{\rm{Take}}\,\,m = 2\,\,\,\, \Rightarrow \,\,\,\,m + 3n = 32\,\,\left( {{\rm{even}}} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = 7\,\, \hfill \cr} \right.$$ 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 • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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