Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 30 Apr 2009
Posts: 108

When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
01 Nov 2009, 13:06
Question Stats:
63% (02:05) correct 37% (02:12) wrong based on 645 sessions
HideShow timer Statistics
When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15? (1) n2 is divisible by 5. (2) t is divisible by 3.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Trying to make CR and RC my strong points
"If you want my advice, Peter," he said at last, "you've made a mistake already. By asking me. By asking anyone. Never ask people. Not about your work. Don't you know what you want? How can you stand it, not to know?" Ayn Rand




Math Expert
Joined: 02 Sep 2009
Posts: 50039

Re: GMAT Prep 2 remainder
[#permalink]
Show Tags
01 Nov 2009, 14:07
When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?(1) n2 is divisible by 5. (2) t is divisible by 3. From the stem: \(n=3p+2\) and \(t=5q+3\). \(nt=15pq+9p+10q+6\), we should find the remainder when this expression is divided by 15. (1) \(n2=5m\) > \(n=5m+2=3p+2\) > \(5m=3p\), \(15m=9p\) > \(nt=15pq+ 9p+10q+6=15pq+ 15m+10q+6\). Clearly \(15pq\) and \(15m\) are divisible by 15, so remainder by dividing these components will be 0. But we still know nothing about \(10q+6\). Not sufficient. (2) t is divisible by 3 means that \(5q+3\) is divisible by 3 > 5q is divisible by 3 or q is divisible by 3 > \(5q=5*3z=15z\) > \(10q=30z\) > \(nt=15pq+9p+ 10q+6=15pq+9p+ 30z+6\). \(15pq\) and \(30z\) are divisible by 15. Know nothing about \(9p+6\). Not sufficient. (1)+(2) \(9p=15m\) and \(10q=30z\) > \(nt=15pq+ 9p+ 10q+6=15pq+15m+30z+6\). Remainder when this expression is divided by 15 is 6. Sufficient. Answer: C. OR:From the stem: \(n=3p+2\) and \(t=5q+3\). (1) n2 is divisible by 5 > \(n2=5m\) > \(n=5m+2\) and \(n=3p+2\) > general formula for \(n\) would be \(n=15k+2\) (about deriving general formula for such problems at: goodproblem90442.html#p723049 and manhattanremainderproblem93752.html#p721341) > \(nt=(15k+2)(5q+3)=15*5kq+15*3k+10q+6\) > first two terms are divisible by 15 (\(15*5kq+15*3k\)) but we don't know about the last two terms (\(10q+6\)). Not sufficient. (2) t is divisible by 3 > \(t=3r\) and \(t=5q+3\) > general formula for \(t\) would be \(t=15x+3\) > \(nt=(3p+2)(15x+3)=15*3px+9p+15*2x+6\). Not sufficient. (1)+(2) \(nt=(15k+2)(15x+3)=15*15kx+15*3k+15*2x+6\) this expression divided by 15 yields remainder of 6. Sufficient. Answer: C.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Intern
Joined: 24 May 2010
Posts: 6
Location: New York, USA

Re: GMAT Prep 2 remainder
[#permalink]
Show Tags
24 May 2010, 17:52
If the explanation above is not helpful, you may find a step by step video solution of this question useful. On GMATFix site, this is GMATPrep question 1045
Best of luck, Patrick



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8405
Location: Pune, India

Re: DS problem : remainders
[#permalink]
Show Tags
28 Nov 2010, 18:36
hogwarts wrote: Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks! When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15? (1) n2 is divisible by 5. (2) t is divisible by 3. The correct answer is (C)  both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks! This is how I would approach this question. When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer) and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.) What is the remainder when the product nt is divided by 15?So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about ( 9a + 10b + 6) yet. Stmnt 1: n2 is divisible by 5. From above, n  2 is just 3a. If n  2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient. Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient. Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient. Answer (C).
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Manager
Status: what we want to do, do it as soon as possible
Joined: 24 May 2010
Posts: 83
Location: Vietnam
WE 1: 5.0

Re: GMAT Prep 2 remainder
[#permalink]
Show Tags
16 Mar 2011, 00:55
Bunuel wrote: When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?
(1) n2 is divisible by 5. (2) t is divisible by 3.
Answer: C. I have another approach to this ds, plz correct me if i'm wrong. n=3x + 2 t = 5y + 3 Clearly we cannot solve the problem with either n or t. We need both information concerning n and t because we need to figure out the remaining of n*t. => left with C or E (1): n2 is divisible by 5 & n=3x + 2 => x is multiple of 5. (2): t is divisible by 3. & t = 5y + 3 => y is multiple of 3 (1)& (2) => n*t = (3x+2) (5y+3) = (3x*5y) + (9x) + (10y) + 6 we know that: x is multiple of 5, y is multiple of 3 so: (3x*5y) + (9x) + (10y) + 6 will have remaining of 6 because: each (3x*5y); (9x); (10y) is all multiple of 15.
_________________
Consider giving me kudos if you find my explanations helpful so i can learn how to express ideas to people more understandable.



Retired Moderator
Joined: 16 Nov 2010
Posts: 1436
Location: United States (IN)
Concentration: Strategy, Technology

Re: GMAT Prep 2 remainder
[#permalink]
Show Tags
17 Mar 2011, 00:34
I tried plugging numbers: n = 3k + 2 = 2,5,8,11,14,17 t = 5m + 3 = 3,8,13,18,23,28 (1) n  2 = 5l => n = 5l + 2 = 2,7,12,17 So n = 15k + 2 = 2,17,32,47 But n* t = 6 (2*3), 16(2*8) so n/15 can have rem 1 or 6 Hence (1) is not enough (2) t = 3p = 3,6,12,15,18,21 So t = 15q + 3 = 3,18,33,48 But n*t = 6, 15, so rem can be 6, 0 etc. Combining (1) and (2), it can be seen that nt = 15 * an integer + 6, so remainder is 6, answer is C.
_________________
Formula of Life > Achievement/Potential = k * Happiness (where k is a constant)
GMAT Club Premium Membership  big benefits and savings



Manager
Status: Quant 50+?
Joined: 02 Feb 2011
Posts: 100
Concentration: Strategy, Finance

Re: GMAT Prep 2 remainder
[#permalink]
Show Tags
02 Oct 2011, 17:42
I did a similar thing with plugging numbers.
First you have to see that (1) and (2) alone are not sufficient alone before it really works in a time effective manner though.
(1) n = 17,32,47,etc, t still has so many values and remainder can differ (17*3 and 17*8 for example) (2) t = 18,33,48,etc, same as above but with n
(1)+(2)
(15p+2)(15p+3) will always have r6 when divided by 15
breaking it out in factored form like that is helpful for me to see it very clearly.



Intern
Joined: 11 Jul 2013
Posts: 33

Re: DS problem : remainders
[#permalink]
Show Tags
19 Oct 2013, 08:00
VeritasPrepKarishma wrote: hogwarts wrote: Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks! When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15? (1) n2 is divisible by 5. (2) t is divisible by 3. The correct answer is (C)  both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks! This is how I would approach this question. When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer) and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.) What is the remainder when the product nt is divided by 15?So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about ( 9a + 10b + 6) yet. Stmnt 1: n2 is divisible by 5. From above, n  2 is just 3a. If n  2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient. Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient. Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient. Answer (C). so from statement 1 we got 10b+6 if b=1 we get 16 then rem =1 if b=2 we got 106 so remainder =1 if b=3 we get 1006 so remainder =1 ..............................so i think a is sufficient .............what am i doing wrong



Math Expert
Joined: 02 Sep 2009
Posts: 50039

Re: DS problem : remainders
[#permalink]
Show Tags
20 Oct 2013, 05:38
tyagigar wrote: VeritasPrepKarishma wrote: hogwarts wrote: Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks! When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15? (1) n2 is divisible by 5. (2) t is divisible by 3. The correct answer is (C)  both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks! This is how I would approach this question. When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer) and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.) What is the remainder when the product nt is divided by 15?So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about ( 9a + 10b + 6) yet. Stmnt 1: n2 is divisible by 5. From above, n  2 is just 3a. If n  2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient. Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient. Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient. Answer (C). so from statement 1 we got 10b+6 if b=1 we get 16 then rem =1 if b=2 we got 106 so remainder =1 if b=3 we get 1006 so remainder =1 ..............................so i think a is sufficient .............what am i doing wrong 10b above means 10*b, 10 multiplied by b. If b=2, then 10b+6=10*2+6=26 not 106; If b=2, then 10b+6=10*3+6=36 not 1006. Does this make sense?
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 31 Oct 2015
Posts: 33

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
11 Nov 2015, 20:57
When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?
n = 3a + 2, t = 5b + 3. Q: (3a + 2)(5b + 3)/15 = c + r/15. What is the value of r?
r is knowable only if it can be determined that each of 3a and 5b are divisible by 15, or in other words that a is divisible by 5 and b is divisible by 3, in which case the remainder would be the only remaining element that is clearly not divisible by 15.
nt = (3a +2)(5b +3) = 3a x 5b + 3a x3 + 10b + 6
if the first three elements of this equation are divisible by 15 the remainder would be 6. If divisibility of any of the variablecontaining elements of this equation cannot be determined then the remainder cannot be determined.
St1: n2 is divisible by 5.
n = 3a + 2 3a = n2
if n2 divisible by 5, then 3a is divisible by 5 (at least one factor of a = 5), and therefore 3a is divisible by 15.
However, divisibility of 5b by 15 cannot be determined, and therefore the value of r cannot be determined.
INSUFF.
(2) t is divisible by 3.
t = 5b + 3 5b + 3 is divisible by 3, therefore 5b is divisible by 3(at least one factor of b is 3) and 5b is divisible by 15.
However, divisibility of 3a by 15 cannot be determined, and therefore the value of r cannot be determined.
INSUFF.
Combined: 3a, and 5b are both divisible by 15, and therefore, all variablecontaining elements of the equation nt = (3a +2)(5b +3) = 3a x 5b + 3a x3 + 10b + 6 are divisible by 15 and the remainder is 6.
SUFF.
Answer is C.



Manager
Joined: 07 May 2015
Posts: 87

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
28 Feb 2016, 10:09
Hi Bunuel, thanks for the solution. I tried to solve statement two and I below is what I am getting. What am i doing wrong? T = 5q+3 (given) T = 3k (statement 2) So 5q + 3 = 3k > 10q + 6 = 6k and this does not assures that 10q+6 is a multiple of 15. What am i missing in solving it this way? Thanks in advance! Bunuel wrote: When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?(1) n2 is divisible by 5. (2) t is divisible by 3. From the stem: \(n=3p+2\) and \(t=5q+3\). \(nt=15pq+9p+10q+6\), we should find the remainder when this expression is divided by 15. (1) \(n2=5m\) > \(n=5m+2=3p+2\) > \(5m=3p\), \(15m=9p\) > \(nt=15pq+ 9p+10q+6=15pq+ 15m+10q+6\). Clearly \(15pq\) and \(15m\) are divisible by 15, so remainder by dividing these components will be 0. But we still know nothing about \(10q+6\). Not sufficient. (2) t is divisible by 3 means that \(5q+3\) is divisible by 3 > 5q is divisible by 3 or q is divisible by 3 > \(5q=5*3z=15z\) > \(10q=30z\) > \(nt=15pq+9p+ 10q+6=15pq+9p+ 30z+6\). \(15pq\) and \(30z\) are divisible by 15. Know nothing about \(9p+6\). Not sufficient. (1)+(2) \(9p=15m\) and \(10q=30z\) > \(nt=15pq+ 9p+ 10q+6=15pq+15m+30z+6\). Remainder when this expression is divided by 15 is 6. Sufficient. Answer: C. OR:From the stem: \(n=3p+2\) and \(t=5q+3\). (1) n2 is divisible by 5 > \(n2=5m\) > \(n=5m+2\) and \(n=3p+2\) > general formula for \(n\) would be \(n=15k+2\) (about deriving general formula for such problems at: goodproblem90442.html#p723049 and manhattanremainderproblem93752.html#p721341) > \(nt=(15k+2)(5q+3)=15*5kq+15*3k+10q+6\) > first two terms are divisible by 15 (\(15*5kq+15*3k\)) but we don't know about the last two terms (\(10q+6\)). Not sufficient. (2) t is divisible by 3 > \(t=3r\) and \(t=5q+3\) > general formula for \(t\) would be \(t=15x+3\) > \(nt=(3p+2)(15x+3)=15*3px+9p+15*2x+6\). Not sufficient. (1)+(2) \(nt=(15k+2)(15x+3)=15*15kx+15*3k+15*2x+6\) this expression divided by 15 yields remainder of 6. Sufficient. Answer: C.



Intern
Joined: 03 Jan 2016
Posts: 4

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
05 May 2016, 20:29
kt00381n wrote: When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?
(1) n2 is divisible by 5. (2) t is divisible by 3. As per the stated, n = 3p + 2 => n2 = 3p i.e. n2 is a multiple of 3. t = 5q + 3 => t3 = 5q i.e. t3 is a multiple of 5. Statement 1 says n2 is a multiple of 5. Hence, n2 is divisible by both 3 and 5. So, it should be divisible by 15. => n2 = 15m => n = 15m +2.... (1) But stat 1 does not say anything about t2. Hence insufficient. (or so u can deduce) Statement 2 says t2 is a multiple of 3. Hence, t3 is divisible by both 3 and 5. So, it should be divisible by 15. => t3 = 15m => t = 15s +3.... (2) But stat 2 does not say anything about n. Hence insufficient. On combining these two statements, nt= 15m*15s + 15m*3+15s*2+3*2 So nt on dividing by 15 gives 6 as the remainer. (All other terms are divisible by 15)



Moderator
Joined: 22 Jun 2014
Posts: 1029
Location: India
Concentration: General Management, Technology
GPA: 2.49
WE: Information Technology (Computer Software)

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
10 Jun 2016, 12:09
kt00381n wrote: When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?
(1) n2 is divisible by 5. (2) t is divisible by 3. n=3a+2 AND t=5b+3nt = 15ab + 10b + 9a + 6  divide it by 15 Nt/15 = ab + (2/3)b + (3/5)a + 6/15. We see if we know both b and a are divisible by 3 and 5 respectively then we remainder is 6 else remainder would be something else depending on the value of a and b.Stmt1:As we know n=3a+2 then n2 = 3a. Stmt1 says n2 is divisible by 5 that is3a is divisible by 5. Hence a is divisible by 5. BUT what about b? INSUFF. Stmt2:As we know t=5b+3 Stmt2 says t is divisible by 3 that is b is divisible by 3. BUT what about a? INSUFF. Combining stmt1 and stmt2 we know a and b are divisible by 5 and 3 respectively. This is what we required to know as mentioned above. Sufficient. Answer is C.
_________________
 Target  720740 http://gmatclub.com/forum/informationonnewgmatesrreportbeta221111.html http://gmatclub.com/forum/listofoneyearfulltimembaprograms222103.html



Director
Joined: 26 Oct 2016
Posts: 642
Location: United States
Concentration: Marketing, International Business
GPA: 4
WE: Education (Education)

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
26 Dec 2016, 07:32
VeritasPrepKarishma wrote: hogwarts wrote: Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks! When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15? (1) n2 is divisible by 5. (2) t is divisible by 3. The correct answer is (C)  both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks! This is how I would approach this question. When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer) and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.) What is the remainder when the product nt is divided by 15?So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about ( 9a + 10b + 6) yet. Stmnt 1: n2 is divisible by 5. From above, n  2 is just 3a. If n  2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient. Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient. Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient. Answer (C). Very straight forward, crisp and compact solution to this problem.
_________________
Thanks & Regards, Anaira Mitch



Intern
Joined: 04 Jan 2017
Posts: 12

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
15 Jan 2017, 01:38
Hi everyone,
Thanks for all the explanations, it´s always extremely helpful.
I got the right answer but it took me 4min 44 sec. Any suggestions to get get the time down.
Thanks



Math Expert
Joined: 02 Sep 2009
Posts: 50039

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
15 Jan 2017, 01:45



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 384

Re: When positive integer n is divided by 3, the remainder is 2
[#permalink]
Show Tags
05 Oct 2018, 09:06
kt00381n wrote: When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?
(1) n2 is divisible by 5. (2) t is divisible by 3.
(The solution presented below is VERY similar to Karishma´s approach.) Obs.: don´t be "frightened" by the manyintegerrepresentingletters below. The MEANING they provide is simple and far more important than the "heavy notation"! \(n,t\,\, \ge \,\,1\,\,{\rm{ints}}\,\,\,\left( * \right)\) \(n = 3M + 2\,\,\,\,\left( {{\mathop{\rm int}} \,\,M \ge 0\,\,{\rm{by}}\,\,\left( * \right)} \right)\) \(t = 5K + 3\,\,\,\,\left( {{\mathop{\rm int}} \,\,K \ge 0\,\,{\rm{by}}\,\,\left( * \right)} \right)\) \(nt = \underbrace {15MK}_{{\text{multiple}}\,{\text{of}}\,\,15} + 9M + 10K + 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\boxed{\,\,?\,\,\,\,:\,\,\,\,{\text{remainder}}\,\,{\text{by}}\,\,15\,\,\,}\) \(\left( 1 \right)\,\,n  2 = 5J\,\,\,\left( {{\mathop{\rm int}} \,\,J \ge 0\,\,{\rm{by}}\,\,\left( * \right)} \right)\,\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {n,t} \right) = \left( {2,3} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( {M,K} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{6}}\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {n,t} \right) = \left( {2,8} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( {M,K} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{1}}\,\, \hfill \cr} \right.\) \(\left( 2 \right)\,\,t = 3G\,\,\,\left( {{\mathop{\rm int}} \,\,G \ge 1\,\,{\rm{by}}\,\,\left( * \right)} \right)\,\,\,\,\,\left\{ \matrix{ \,\left( {{\rm{Re}}} \right){\rm{Take}}\,\,\left( {n,t} \right) = \left( {2,3} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( {M,K} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{6}}\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {n,t} \right) = \left( {5,3} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( {M,K} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{0}}\,\, \hfill \cr} \right.\) \(\left( {1 + 2} \right)\,\,\) \(\left\{ \matrix{ \,\,n  2 = 3M \hfill \cr \,\,n  2 = 5J \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,n  2 = 15H\,\,\,\left( {{\mathop{\rm int}} \,\,H \ge 0\,\,{\rm{by}}\,\,\left( * \right)} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,n = 15H + 2\) \(\left\{ \matrix{ \,\,t  3 = 5K \hfill \cr \,\,t = 3G\,\,\,\, \Rightarrow \,\,\,t  3 = 3W \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,t  3 = 15D\,\,\,\left( {{\mathop{\rm int}} \,\,D \ge 0\,\,{\rm{by}}\,\,\left( * \right)} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,t = 15D + 3\) \(\left\{ \matrix{ n = 15H + 2 \hfill \cr t = 15D + 3 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,?\,\,\,:\,\,\,\,\,\,nt = 15\,\underbrace {\left[ {15HD + 3H + 2D} \right]}_{{\mathop{\rm int}} } + 6\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = 6\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\) This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio.
_________________
Fabio Skilnik :: https://www.GMATH.net (Math for the GMAT) Course release PROMO : finish our test drive till 31/Oct with (at least) 60 correct answers out of 92 (12questions Mock included) to gain a 60% discount!




Re: When positive integer n is divided by 3, the remainder is 2 &nbs
[#permalink]
05 Oct 2018, 09:06






