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# when a positive integer x is divided by 3, the remainder is

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Senior Manager
Joined: 29 Aug 2005
Posts: 262
when a positive integer x is divided by 3, the remainder is [#permalink]

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13 Jun 2008, 00:00
1
when a positive integer x is divided by 3, the remainder is 1, when positive integer y is divided by 5 , the remainder is 1. If x+y is divisible by 5 what is the least possible value of y

1
6
7
11
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this is how i solve such questions (it takes me 1-1.5 mins to solve this)
my point is that is this approach an optimal one (if no then please suggest an optimal approach)

X =3N+1
Y=5N+1
X+Y=15N+16
16/15 remainder is 1 therefore the least value that y could have is 1

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Joined: 12 Jun 2008
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Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 01:51
vdhawan1 wrote:
when a positive integer x is divided by 3, the remainder is 1, when positive integer y is divided by 5 , the remainder is 1. If x+y is divisible by 5 what is the least possible value of y

1
6
7
11
13

this is how i solve such questions (it takes me 1-1.5 mins to solve this)
my point is that is this approach an optimal one (if no then please suggest an optimal approach)

X =3N+1
Y=5N+1
X+Y=15N+16
16/15 remainder is 1 therefore the least value that y could have is 1

I don't get why you assume that the 'N' in X=3N+1 and Y=5N+1 is the same (there is no reason for that). And I don't understand what you mean after that either (why do you divide 16 by 15 ? How does it translate into Y least value is 1 ?)

Correct answer is still 1 : just try X=4 and Y=1 and you can verify it works.
Director
Joined: 27 May 2008
Posts: 531
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 01:57
how can y = 1, when question says y = 5N+1, N = 1,2,3 .... N can not be zero.

x = 4, y = 11
x+y = 15, divisible by 5.
Current Student
Joined: 12 Jun 2008
Posts: 287
Schools: INSEAD Class of July '10
Re: when a positive integer x is divided by [#permalink]

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Updated on: 13 Jun 2008, 03:17
durgesh79 wrote:
how can y = 1, when question says y = 5N+1, N = 1,2,3 .... N can not be zero.

x = 4, y = 11
x+y = 15, divisible by 5.

Why do you assume N cannot be 0 ? I don't get it.

1 is a positive integer whose remainder is 1 when divided by 5 (since 1=5*0+1)

Edit : and even if you assume (for no reason) that N should be positive, then the answer would be 6 and not 11. Try x=4 and y=6, you'll see

Originally posted by Oski on 13 Jun 2008, 02:06.
Last edited by Oski on 13 Jun 2008, 03:17, edited 1 time in total.
Director
Joined: 27 May 2008
Posts: 531
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 02:23
Oski wrote:
Edit : and even if you assume (for no reason) that N should be positive, then the answer would be 6 and not 11. Try x=4 and y=6, you'll see

oops... yes the answer should be 6.

Regarding the point why N can not be zero, I thought when we talk of y divided by x and remainder r, it is assumed that y>x, so that the integer quotient is greater than or equal to 1. Please correct me if I'm wrong.
Senior Manager
Joined: 29 Aug 2005
Posts: 262
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 02:32
Oh k

the OA for this one is A
_________________

The world is continuous, but the mind is discrete

Current Student
Joined: 12 Jun 2008
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Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 03:20
durgesh79 wrote:
Regarding the point why N can not be zero, I thought when we talk of y divided by x and remainder r, it is assumed that y>x, so that the integer quotient is greater than or equal to 1. Please correct me if I'm wrong.

I see no reason for assuming that.

For me -72 is divisible by 9 even if -72 is not greater than 9. Plus 0 is divisible by any integer (and that would not be the case if we make your assumption)
Director
Joined: 27 May 2008
Posts: 531
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 06:55
Oski wrote:
durgesh79 wrote:
Regarding the point why N can not be zero, I thought when we talk of y divided by x and remainder r, it is assumed that y>x, so that the integer quotient is greater than or equal to 1. Please correct me if I'm wrong.

I see no reason for assuming that.

For me -72 is divisible by 9 even if -72 is not greater than 9. Plus 0 is divisible by any integer (and that would not be the case if we make your assumption)

My assumption was only in the case of +ve remainders, but i guess the assumption is wrong. thanks for pointing it out.

Remainder rule from today : if y is divided by x and the remainder is r, then y = Nx + r, where N = 0,1,2,3........
Current Student
Joined: 12 Jun 2008
Posts: 287
Schools: INSEAD Class of July '10
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 06:58
durgesh79 wrote:
Remainder rule from today : if y is divided by x and the remainder is r, then y = Nx + r, where N = 0,1,2,3........

Not in this particular question, but in general N can even be negative.

General rule would be : if y is divided by x and the remainder is r, then y = Nx + r, where N is integer
SVP
Joined: 29 Mar 2007
Posts: 2493
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 06:59
vdhawan1 wrote:
when a positive integer x is divided by 3, the remainder is 1, when positive integer y is divided by 5 , the remainder is 1. If x+y is divisible by 5 what is the least possible value of y

1
6
7
11
13

this is how i solve such questions (it takes me 1-1.5 mins to solve this)
my point is that is this approach an optimal one (if no then please suggest an optimal approach)

X =3N+1
Y=5N+1
X+Y=15N+16
16/15 remainder is 1 therefore the least value that y could have is 1

That was pretty simple.

1/5 --> remainder is 1.

A
Current Student
Joined: 12 Jun 2008
Posts: 287
Schools: INSEAD Class of July '10
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 07:01
GMATBLACKBELT wrote:
That was pretty simple.

1/5 --> remainder is 1.

A

This is not a proof since it misses every properties of x (which has some).
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Joined: 30 Apr 2008
Posts: 1841
Location: Oklahoma City
Schools: Hard Knocks
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 07:02
This is a good question. I wasn't even thinking of 0 as an option. I immediately started out with x=4, y = 6, and when 4 + 6 = 10/5 = 2, then I thought y must = 6, but x = 4, divided by 3, is 1 w/ remainder 1. y= 1, 1/5 = 0 w/remainder 1, and 1 + 4 =5, obviously divisible by 5.

i guess the point of this question is we must think about 0 as a number and not just go for integers unless the question stem says so.

vdhawan1 wrote:
Oh k

the OA for this one is A

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Current Student
Joined: 12 Jun 2008
Posts: 287
Schools: INSEAD Class of July '10
Re: when a positive integer x is divided by [#permalink]

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13 Jun 2008, 07:08
jallenmorris wrote:
huh?

I'm just saying this is not a proof that the answer is (1). You also have to show that it works for at least one value of x that verifies the specified properties.
VP
Joined: 21 Jul 2006
Posts: 1455
Re: when a positive integer x is divided by [#permalink]

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17 Sep 2008, 18:18
durgesh79 wrote:
Oski wrote:
Edit : and even if you assume (for no reason) that N should be positive, then the answer would be 6 and not 11. Try x=4 and y=6, you'll see

oops... yes the answer should be 6.

Regarding the point why N can not be zero, I thought when we talk of y divided by x and remainder r, it is assumed that y>x, so that the integer quotient is greater than or equal to 1. Please correct me if I'm wrong.

I don't agree. For example, when you divide 3 by 5, the remainder is 3. On the other hand, when you divide 8 by 5, the remainder is also 3. So the numerator doesn't always have to be bigger than the denominator.
SVP
Joined: 29 Aug 2007
Posts: 2427
Re: when a positive integer x is divided by [#permalink]

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17 Sep 2008, 21:45
tarek99 wrote:
GMATBLACKBELT wrote:
vdhawan1 wrote:
when a positive integer x is divided by 3, the remainder is 1, when positive integer y is divided by 5 , the remainder is 1. If x+y is divisible by 5 what is the least possible value of y

1
6
7
11
13

this is how i solve such questions (it takes me 1-1.5 mins to solve this)
my point is that is this approach an optimal one (if no then please suggest an optimal approach)

X =3N+1
Y=5N+1
X+Y=15N+16

am i missing something here? shouldn't 3N+5N equal to 8N? and 1+1=2? where from the 16??

16/15 remainder is 1 therefore the least value that y could have is 1

That was pretty simple.

1/5 --> remainder is 1.

A

Thats not a valid reasoning. The same quotient, N, may not be true for both x and y when divided by 3 and 5 respectively.

x = 3n + 1
y = 5p + 1

since (x+y) is divisible by 5, x+y can be as small as 5 but not 0 or smaller than 0 because x and y each is a +ve integer. so our concern is the smallest value for y. if that is the case, what is the possible smallest value for y? first it is 1, then 2 and so on. lets see whether 1 is possible value for y.

y's value depends on x's value. since x + y is divisible by 5, x should be 5k-1 to have y's value 1. the possible value for x = 5k-1 are 4, 9, 19, 29, 39, 49 etc. but 9, 29, 39 cannot be the values of x cuz they do not yeild 1 as reminder if they each are divided by 3. so the possible values are 4 or 19, 49 etc. in each case y is 1. so this is the smallest possible value for y.

//A//
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VP
Joined: 17 Jun 2008
Posts: 1479
Re: when a positive integer x is divided by [#permalink]

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17 Sep 2008, 23:23
This is how I approaced.

x+y = 5k for k = 0,1,2,3,.....

or y = 5k -x
or y = 5k -(3m+1) for m = 0,1,2,3,......

Now, trying substitution,
k = 0, m = 0, y is -ve....not possible.
k=1, m=1, y = 1.

Hence, the smallest positive value of y will be 1.
VP
Joined: 21 Jul 2006
Posts: 1455
Re: when a positive integer x is divided by [#permalink]

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18 Sep 2008, 05:43
GMAT TIGER,

how did you get x to be 5k-1? Also, i'm trying to understand this problem abstractly because I feel that this can be solved without doing any math. First of all, I remember I read from the Manhattan GMAT book "Number Properties" that when x and y are divisible by, for example, 5, then both x+y and x-y will also be divisible by 5.

However, in this problem, y is definitely not divisible by 5 otherwise the remainder wouldn't be 1. And we don't know much about x either. So how can we go about this abstractly? if you guys know what I mean? After all, this is suppose to be a GMAT question, so there is always an easy and sneaky way to reach to an answer with almost minimal effort.
VP
Joined: 21 Jul 2006
Posts: 1455
Re: when a positive integer x is divided by [#permalink]

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18 Sep 2008, 06:56
I found a quick approach!!!

Like I said before, if a is divisible by 5 and if a= x+y, then it means that both x and y separately should be divisible by 5. However, sometimes a can be divisible by 5, but x and y may not necessarily be divisible by 5. In that case, the sum of the remainders of x and y should be a multiple of 5., hence divisible by 5.

We already know that when y is divided by 5, the remainder is 1. So when x is divided by 5, x's remainder should add up to y's remainder of 1 to give us 5, hence, x should give us a remainder of 4.

With this approach, the value of y automatically becomes 1, 6, 11, or etc....the least value is obviously 1!

This is how you're pretty much expected to approach the math section of the GMAT. If you follow this approach, you can answer this type of question literally in less than 10 seconds!

Hope this can make your life MUCH EASIER!
Senior Manager
Joined: 04 Aug 2008
Posts: 363
Re: when a positive integer x is divided by [#permalink]

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07 Nov 2008, 14:12
Can someone explain how is the answer 1 or A

The question says that Y divided with 5 has a remainder 1
that means Y should be at least 6 to produce remainder 1
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Manager
Joined: 30 Sep 2008
Posts: 111
Re: when a positive integer x is divided by [#permalink]

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07 Nov 2008, 18:23
spiridon wrote:
Can someone explain how is the answer 1 or A

The question says that Y divided with 5 has a remainder 1
that means Y should be at least 6 to produce remainder 1

1 / 5 = 0, remander 1

Keep this in your mind and don't assume (Y should be at least 6 to produce remainder 1)

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This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

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Re: when a positive integer x is divided by   [#permalink] 07 Nov 2008, 18:23
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# when a positive integer x is divided by 3, the remainder is

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