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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 4 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: 2413 messages Followed by: 18 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 The smallest positive integer that is divisible by both 4 and 14 is their LCM, which is 28. Thus 28 + 1 = 29 will yield a remainder of 1 when divided by both 4 and 14. We can add 28 to each previous dividend to obtain additional dividends: 29 + 28 = 57 57 + 28 = 85 Since the next number will be 85 + 28 = 113, which is a 3-digit number, we can stop at 85. Thus, we have three 2-digit numbers (29, 57, and 85) that will yield a remainder of 1 when divided by both 4 and 14. Answer: A _________________ 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 25 May 2010 Posted: 15250 messages Followed by: 1861 members Upvotes: 13060 GMAT Score: 790 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 Let x = the two-digit number. Since dividing by 4 leaves a remainder of 1, x is equal to one more than a multiple of 4: x = 4k + 1 Since dividing by 14 leaves a remainder of 1, x is equal to one more than a multiple of 14: x = 14m + 1 Since the two expressions in blue must be equal, we get: 4a + 1 = 14b + 1 4a = 14b 2a = 7b a = (7/2)b If b=2, then a=7. If b=4, then a=14. If b=6, then a=21. The resulting values for a indicate that a must be a POSITIVE MULTIPLE OF 7. 21 is the largest multiple of 7 such that 4a+1 will yield a two-digit integer. Thus, there are 3 options for a: 7, 14, 21. Since there are 3 options for a, there are 3 possible values for x. The correct answer is A. _________________ 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 08 Dec 2008 Posted: 12821 messages Followed by: 1247 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 OA A Source: Veritas Prep 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 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! ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 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 $?\,\,\,\,:\,\,\,\,\# \,\,N\,\,,\,\,\,10 \leqslant N \leqslant 99\,\,\,{\text{with}}\,\,\,\left( * \right)$ $\left( * \right)\,\,\,\,\left. \begin{gathered} N = 4M + 1\,\,\,\,\left( {M \geqslant 3\,\,\operatorname{int} } \right)\,\,\,\,\,\, \hfill \\ N = 14K + 1\,\,\,\left( {K \geqslant 1\,\,\operatorname{int} } \right) \hfill \\ \end{gathered} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,N - 1\,\,{\text{multiple}}\,\,{\text{of}}\,\,LCM\left( {4,14} \right) = 28$ $\left\{ \begin{gathered} \,\,10 \leqslant N \leqslant 99 \hfill \\ \,\,N = 28L + 1\,\,\,\,\left( {L \geqslant 1\,\,\operatorname{int} } \right) \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,L = 1,2\,\,{\text{or}}\,\,3\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 3$ This solution follows the notations and rationale taught in the GMATH method. 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