1) When positive integer n is divided by 3, remainder is 2. When positive integer t is divided by 5, remainder is 3. What is the remainder when product nt is divided by 15.
a) n 2 is divisible by 5
b) t is divisible by 3
please provide explaination.
GMAT Prep Question
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Q. When positive integer n is divided by 3, remainder is 2. When positive integer t is divided by 5, remainder is 3. What is the remainder when product nt is divided by 15.
n can be 5 8 11 14 17 20 23 26 29 32 and so on
t can be 8 13 18 23 28 33 38 43 48 53 and so on
Each option alone is not suff since one speaks about n and other about t
Both together...
n 2 is divisible by 5 and t is divisible by 3
again, n can be 17 32 47 and so on
and t can be 18 48 and so on
again the remainder will be diff every time
Ans E.
Sorry....i didnt calculate properly.....the remainder is 6 everytime.....
Sorry guys... ;(
ans C.
n can be 5 8 11 14 17 20 23 26 29 32 and so on
t can be 8 13 18 23 28 33 38 43 48 53 and so on
Each option alone is not suff since one speaks about n and other about t
Both together...
n 2 is divisible by 5 and t is divisible by 3
again, n can be 17 32 47 and so on
and t can be 18 48 and so on
again the remainder will be diff every time
Ans E.
Sorry....i didnt calculate properly.....the remainder is 6 everytime.....
Sorry guys... ;(
ans C.
Last edited by Suyog on Fri Dec 21, 2007 12:08 pm, edited 1 time in total.

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1) When positive integer n is divided by 3, remainder is 2. When positive integer t is divided by 5, remainder is 3. What is the remainder when product nt is divided by 15.
i.e n = 3a + 2 & t =5b+3
a) n 2 is divisible by 5
i.e n =5c + 2 also n =3a +2
so 5c +2 = 3a +2
or 5c = 3a
so here we can see that the values of a & c must be such that 5c= 3a
i.e c should be a multiple of 3 & a of 5
e.g c = 3 & a =5 i.e so n can be 17, 32 etc
& t = 5b+3 where b can be any value from 1,2,3........
so the reaminder will be diff each time INSUFF
b) t is divisible by 3
i.e t =3d also t =5b+3
so 3d = 5b +3  (eqn 1)
also n = 3a+2 so nt =(3a+2)(3d),
we cannot determine the remainder here coz a can assume any value here INSUFF
COMBINE:
now we from 1 that n= 5c +2
& from 2, t =3d
so nt = (5c +2)(3d) = 15cd +6d
now when we dicvide this by 15 the factor 15cd will get divided evenly so reaminder will be produced by the divison 6d/15
now from eqn 1 we have
3d =5b+3 or
d =(5b + 3)/3
since d is an inetger so 5b+3 must be a multiple of 3
i.e it can be 18, 33......
so 6d = 18*6, 33*6......
in each case the remainder will be 3
C SUFF
in each case when we divide
i.e n = 3a + 2 & t =5b+3
a) n 2 is divisible by 5
i.e n =5c + 2 also n =3a +2
so 5c +2 = 3a +2
or 5c = 3a
so here we can see that the values of a & c must be such that 5c= 3a
i.e c should be a multiple of 3 & a of 5
e.g c = 3 & a =5 i.e so n can be 17, 32 etc
& t = 5b+3 where b can be any value from 1,2,3........
so the reaminder will be diff each time INSUFF
b) t is divisible by 3
i.e t =3d also t =5b+3
so 3d = 5b +3  (eqn 1)
also n = 3a+2 so nt =(3a+2)(3d),
we cannot determine the remainder here coz a can assume any value here INSUFF
COMBINE:
now we from 1 that n= 5c +2
& from 2, t =3d
so nt = (5c +2)(3d) = 15cd +6d
now when we dicvide this by 15 the factor 15cd will get divided evenly so reaminder will be produced by the divison 6d/15
now from eqn 1 we have
3d =5b+3 or
d =(5b + 3)/3
since d is an inetger so 5b+3 must be a multiple of 3
i.e it can be 18, 33......
so 6d = 18*6, 33*6......
in each case the remainder will be 3
C SUFF
in each case when we divide
Regards
Samir
Samir

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This is how I solved it.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 3 implies p is divisible by 3 But we don't know aboyt q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 But we don't know aboyt p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 3 implies p is divisible by 3 But we don't know aboyt q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 But we don't know aboyt p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.

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(Correcting typos in my earlier post)
This is how I solved it.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 5 implies p is divisible by 3 But we don't know about q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 But we don't know about p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.
This is how I solved it.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 5 implies p is divisible by 3 But we don't know about q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 But we don't know about p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.

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(Correcting typos AGAIN!!! in my earlier post)
This is how I solved it.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 5 implies p is divisible by 5 which implies second term is divisible by 15. But we don't know about q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 which implies 10q is divisible by 15 But we don't know about p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.
Calista (Sorry, I should check more crefully for typos before posting).
This is how I solved it.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 5 implies p is divisible by 5 which implies second term is divisible by 15. But we don't know about q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 which implies 10q is divisible by 15 But we don't know about p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.
Calista (Sorry, I should check more crefully for typos before posting).

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HiTSonam wrote:1) When positive integer n is divided by 3, remainder is 2. When positive integer t is divided by 5, remainder is 3. What is the remainder when product nt is divided by 15.
a) n 2 is divisible by 5
b) t is divisible by 3
please provide explaination.
My two cents...
from Q:
n=3i + 2 [i>=0]
t=5j + 3 [j>=0]
so
nt = (3i+2)(5j+3)
=15ij + 9i + 10j + 6
Since the divisor is 15 then 15ij portion can be removed and we consider
Remainder = 9i + 10j + 6
Now
1 & 2 can't be considered for seperately sufficient because each gives infor about either i or j sepearately which is not good enuf.
then consider two together
1. n  2 is divisible by 5
=> 3i is divisible by 5
=> i=0,5,15,20,25....
We cant simply say for all i's the term 9i is divisible by 15.
2. t is divisble by 3
=> 5j + 3 divisible by 3
=> 5j divisible by 3
=> j=0,3,6,9,12.... etc
=> A simple viewing at remainder will show that for all j's 10j term is divisible by 15.
3) Now combine 1 & 2
t(n2) is divisible by 3*15
tn  2t is divisible by 15
tn remainder is inserted here
=> (9i + 10j + 6)2(5j+3) is divisible ...
=> 9i is divisible by 15
so based on 3 and 2 we can say that both terms 9i and 10j in 9i + 10j + 6 are divisible by 15.
Which means 6 is the remainder always and (C) is the answer.
Pardon my being too simplistic at someplaces.
Rgds
Sandy
Decide and do it !!!
I think StarDust845 solution for this problem is superb. Great thinking and superb logic SD845.(Correcting typos AGAIN!!! in my earlier post)
This is how I solved it.
n = 3p + 2 for all p GE 0 (GE means greater than or equal to)
t = 5q + 3 for all q GE 0
nt = 15pq + 9p + 10q + 6  (1)
1) n  2 is divisible by 5 implies p is divisible by 5 which implies second term is divisible by 15. But we don't know about q in (1) above. so INSUFF.
2) t is divisible by 3 implies q is divisible by 3 which implies 10q is divisible by 15 But we don't know about p in (1) above so INSUFF.
Combining both we see that all the first 3 terms in (1) get divided by 15. Reminder is 6. Hence we know that nt is not divisible by 15. Hence C.
Calista (Sorry, I should check more crefully for typos before posting).