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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 ## How many positive two-digit numbers yield a remainder of 1 ##### This topic has 3 expert replies and 0 member replies ### Top Member ## How many positive two-digit numbers yield a remainder of 1 ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14? A. 3 B. 4 C. 5 D. 6 E. 7 OA A Source: Veritas Prep ### GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2012 messages Followed by: 15 members Upvotes: 43 Top Reply BTGmoderatorDC wrote: How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14? A. 3 B. 4 C. 5 D. 6 E. 7 We can use the fact that if a number n leaves a remainder of r when divided by a and b (a ≠ b), then n leaves a remainder of r when divided by the least common multiple (LCM) of a and b. Since the LCM of 4 and 14 is 28, the smallest such two-digit number is 29 (notice that 29/4 = 7 R 1 and 29/14 = 2 R 1). The next two-digit number is 29 + LCM = 29 + 28 = 57. The next one is 57 + 28 = 85. Since the next one will be more than 100, we can stop at 85. Therefore, we see that there are 3 such integers. Answer: A _________________ Scott Woodbury-Stewart Founder and CEO ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1295 messages Followed by: 29 members Upvotes: 59 Top Reply BTGmoderatorDC wrote: How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14? A. 3 B. 4 C. 5 D. 6 E. 7 Source: Veritas Prep Let´s use some powerful "divisibility tools" carefully explained in our course. (Our students perform all operations below in approximately 2min with no rush!) $N = 4Q + 1 = 14J + 1\,\,\,\,\,\,\,\,\left( {Q,J\,\,{\text{ints}}} \right)$ $\left. \begin{gathered} N - 1 = 4Q \hfill \\ N - 1 = 14J\, \hfill \\ \end{gathered} \right\}\,\,\,\,\mathop \Rightarrow \limits^{LCM\,\left( {4,14} \right)\, = \,\,{2^2} \cdot 7} N - 1 = 28K\,\,\,\,\, \Rightarrow \,\,\,\,N = 28K + 1\,\,,\,\,\,K\,\,\operatorname{int}$ $10 \leqslant N \leqslant 99\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,10 \leqslant 28K + 1 \leqslant 99\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,9 \leqslant 28K \leqslant 98$ $9 \leqslant 28K \leqslant 98\,\,\,\, \Rightarrow \,\,\,\,1 \cdot 28 \leqslant 28K \leqslant 3 \cdot 28\,\,\,\,\, \Rightarrow \,\,\,\,1 \leqslant K \leqslant 3\,\,\,\,\, \Rightarrow \,\,\,\,? = 3\,\,\,\,\,\,\,$ 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 08 Dec 2008 Posted: 12663 messages Followed by: 1246 members Upvotes: 5254 GMAT Score: 770 BTGmoderatorDC wrote: How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14? A. 3 B. 4 C. 5 D. 6 E. 7 When it comes to remainders, we have a nice rule that says: If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc. For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc. Two-digit number yields a remainder of 1 when divided by 14. So, the possible values are: 15, 29, 43, 57, 71, 85 and 99 At this point, we have 7 possible values Two-digit number yields a remainder of 1 when divided by 4. Examine each of the 7 values and determine which ones yield a remainder of 1 when divided by 4 They are: 15, 29, 43, 57, 71, 85 and 99 So, there are 3 values that satisfy BOTH conditions. Answer: A 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! • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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