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# If a and b are positive integers such that a/b=2.86, which

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Joined: 18 Aug 2013
Posts: 13
Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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Updated on: 23 Sep 2013, 01:29
You're doing everything correct. I did the same method, then got stuck near the end as you did. Here's how you would finish the problem.

Since b = 50x, and R = 43x, and a/b = 2 + R/b

a = 2b + R.

So a is equal to 143 (x = 1), 283 (x = 2), etc... In each of these, a is a multiple of 11 and 13.

Looking back, Bunuel's method is method is much easier, as it ignores calculations involving the remainder.

Originally posted by grant1377 on 22 Sep 2013, 10:14.
Last edited by grant1377 on 23 Sep 2013, 01:29, edited 1 time in total.
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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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23 Sep 2013, 00:57
grant1377 wrote:
You're doing everything correct. I did the same method, then got suck near the end as you did. Here's how you would finish the problem.

Since b = 50x, and R = 43x, and a/b = 2 + R/b

a = 2b + R.

So a is equal to 143 (x = 1), 283 (x = 2), etc... In each of these, a is a multiple of 11 and 13.

Looking back, Bunuel's method is method is much easier, as it ignores calculations involving the remainder.

Hi mate, thanks for the reply. I got a question.
if x = 1, then 50x would be 50 and 43x would be 43, hence 93.
Can you explain in more detail please?
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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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23 Sep 2013, 01:18
Skag55 wrote:
grant1377 wrote:
You're doing everything correct. I did the same method, then got suck near the end as you did. Here's how you would finish the problem.

Since b = 50x, and R = 43x, and a/b = 2 + R/b

a = 2b + R.

So a is equal to 143 (x = 1), 283 (x = 2), etc... In each of these, a is a multiple of 11 and 13.

Looking back, Bunuel's method is method is much easier, as it ignores calculations involving the remainder.

Hi mate, thanks for the reply. I got a question.
if x = 1, then 50x would be 50 and 43x would be 43, hence 93.
Can you explain in more detail please?

It's 2B, not B --> A = 2B + R = 2*50x + 43x = 143x.
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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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03 Apr 2014, 03:48
$$\frac{a}{b} = 2.86$$

$$\frac{a}{b} = \frac{286}{100}$$

$$\frac{a}{b} = \frac{143}{50}$$

$$a = \frac{143}{50} * b$$

We require to find from the available 5 options that must be a divisor of a"

This also means which of the following 5 options can divide 143 evenly

(Ignore b as its value unknown; Ignore 50 as in denominator)

Only option B = 13 best fits in

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If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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19 Jun 2014, 12:46
Hi,

Sorry for bumping this old thread but i also faced the same question in one of my CATs.
My question is that we have been given a fraction and in the form of a/b..
we are asked to identify the divisor of 'a' , which in this case refers to 'b'
we have a = 2.86*b... Now, since a must be an integer, the only value that satisfies the equation is 50.

Is this question really ambiguous or is it just me not being able to differentiate between factors and divisors...

Thanks,
Gaurav
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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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19 Jun 2014, 13:00
gaurav245 wrote:
Hi,

Sorry for bumping this old thread but i also faced the same question in one of my CATs.
My question is that we have been given a fraction and in the form of a/b..
we are asked to identify the divisor of 'a' , which in this case refers to 'b'
we have a = 2.86*b... Now, since a must be an integer, the only value that satisfies the equation is 50.

Is this question really ambiguous or is it just me not being able to differentiate between factors and divisors...

Thanks,
Gaurav

A factor and a divisor are the same thing (well, we can say that a factor is a positive divisor but this is not relevant here). We need to find which of the options could be a factor (divisor) of a, not the value of b, which would be impossible to find from a/b = 2.86.
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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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01 Oct 2014, 08:47
M3tm4n wrote:
I solved it with using prime factorization within 20 sec.
286 has primes 2, 11, 13
Maybe it is the wrong or maybe not the best way to solve this.

It's a good way to solve it - just note that 2 does NOT have to be a factor, so if 2 was one of the options on the list then this method wouldn't work.

Simplify the expression first, to 286/100 = 143/50 = 13*11/(5*5*2) and then you see that 13,11 are the only two NECESSARY options.
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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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10 Oct 2014, 08:39
xianster wrote:
If a and b are positive integers such that a/b=2.86, which of the following must be a divisor of a?

A. 10
B.13
C. 18
D. 26
E. 50

Hope to hear from u guys soon! Thanks!

We know that
a

b
= 2.86.
Because 2.86 is a terminating decimal and a and b are integers, it might be easier to convert 2.86 to a fraction:

a

b
= 2.86
a

b
=
286

100
Now, we must reduce:

a

b
=
143

50

It might be easier to think through the problem if we cross multiply:

50 × a = 143 × b

What does that tell us about a and b? Well, we know that 50, a, 143, and b are all integers. Thus both sides of the equation will be integers (the same integer). For that to be true, both sides of the equation must have IDENTICAL prime factorizations.

We know that the left side of the equation has a 2 and 2 5’s in its prime factorization (50 = 5×5×2). Therefore, b must have at least a 2, a 5 and another 5 in its prime factorization. So b is divisible by 50. Furthermore, we know that the right side of the equation has an 11 and a 13 in its prime factorization (143 = 11×13). Therefore, a must have at least an 11 and a 13 in its prime factorization. So a is divisible by 11, 13, and 143.

The question asks about a. We know that a must be divisible by 13.

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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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07 Apr 2015, 15:38
Bunuel wrote:
GMATmission wrote:
GMATmission wrote:
I did it like this:

We know that a/b =2.86, which means the decimal part i.e. 0.86 = Remainder/Divisor
Simplifying the equation, we get Remainder/Divisor = 43/50. So the divisor should be a multiple of 50.Hence answer is E.

Where am I going wrong?

Can experts please comment on where I am going wrong?

You did everything right except that 50 must be a factor of b, which is a divisor in our case, but we are asked about a not b.

So the highlighted part does say that 50 is a divisor, also as per, 0.86 = Remainder/Divisor then how does it not affect 'a'?
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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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07 Apr 2015, 16:38
3
Hi earnit,

The prompt gives us a couple of facts to work with:
1) A and B are positive INTEGERS
2) A/B = 2.86

We can use these facts to figure out POSSIBLE values of A and B. The prompt asks us for what MUST be a divisor of A. Since we're dealing with a fraction, A and B could be an infinite number of different integers, so we have to make both as SMALL as possible; in doing so, we'll be able to find the divisors that ALWAYS divide in (and eliminate the divisors that only SOMETIMES divide in).

The simplest place to start is with...
A = 286
B = 100
286/100 = 2.86

These values are NOT the smallest possible values though (since they're both even, we can divide both by 2)...

A = 143
B = 50
143/50 = 2.86

There is no other way to reduce this fraction, so A must be a multiple of 143 and B must be an equivalent multiple of 50. At this point though, the value of B is irrelevant to the question. We're asked for what MUST divide into A....

Since A is a multiple of 143, we have to 'factor-down' 143. This gives us (11)(13). So BOTH of those integers MUST be factors of A. You'll find the match in the answer choices.

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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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07 Apr 2015, 17:00
EMPOWERgmatRichC wrote:
Hi earnit,

The prompt gives us a couple of facts to work with:
1) A and B are positive INTEGERS
2) A/B = 2.86

We can use these facts to figure out POSSIBLE values of A and B. The prompt asks us for what MUST be a divisor of A. Since we're dealing with a fraction, A and B could be an infinite number of different integers, so we have to make both as SMALL as possible; in doing so, we'll be able to find the divisors that ALWAYS divide in (and eliminate the divisors that only SOMETIMES divide in).

The simplest place to start is with...
A = 286
B = 100
286/100 = 2.86

These values are NOT the smallest possible values though (since they're both even, we can divide both by 2)...

A = 143
B = 50
143/50 = 2.86

There is no other way to reduce this fraction, so A must be a multiple of 143 and B must be an equivalent multiple of 50. At this point though, the value of B is irrelevant to the question. We're asked for what MUST divide into A....

Since A is a multiple of 143, we have to 'factor-down' 143. This gives us (11)(13). So BOTH of those integers MUST be factors of A. You'll find the match in the answer choices.

GMAT assassins aren't born, they're made,
Rich

Right. I got this method.
But what is exactly wrong in the approach above: that states

$$Dividend/Divisor$$ =$$Quotient$$ + $$Remainder/Divisor$$

Ex: 5/2 = 2.5 = 2 + 0.5 => 2 + 1/2 (where 1 is the remainder and 2 is the divisor)
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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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07 Apr 2015, 17:08
Hi earnit,

In real basic terms, you're doing math that has nothing to do with the question that is asked.

In this prompt, the ONLY reason that the "B" variable is there is to help you figure out what the minimum value of the "A" variable is.

Once we know that A is a multiple of 143, and the question asks what MUST be a factor of A, the value of B is no longer needed.

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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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16 Apr 2015, 19:09
2
1
Hi All,

The prompt gives us a couple of facts to work with:
1) A and B are positive INTEGERS
2) A/B = 2.86

We can use these facts to figure out POSSIBLE values of A and B. The prompt asks us for what MUST be a divisor of A. Since we're dealing with a fraction, A and B could be an infinite number of different integers, so we have to make both as SMALL as possible; in doing so, we'll be able to find the divisors that ALWAYS divide in (and eliminate the divisors that only SOMETIMES divide in).

The simplest place to start is with...
A = 286
B = 100
286/100 = 2.86

These values are NOT the smallest possible values though (since they're both even, we can divide both by 2)...

A = 143
B = 50
143/50 = 2.86

There is no other way to reduce this fraction, so A must be a multiple of 143 and B must be an equivalent multiple of 50. At this point though, the value of B is irrelevant to the question. We're asked for what MUST divide into A....

Since A is a multiple of 143, we have to 'factor-down' 143. This gives us (11)(13). So BOTH of those integers MUST be factors of A. You'll find the match in the answer choices.

GMAT assassins aren't born, they're made,
Rich
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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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26 Dec 2015, 10:11
oh god, did not practice remainders for a while..took me some time to think how to tackle..
ok, so a=2b+r/b. where r is the remainder.
r/b=0.86 or 86/100 or 43/50.
now, suppose b=50, then a is 143. this is a multiple of 13. so the answer is B.
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If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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26 Mar 2016, 16:53
1
1
in my oppinion I think that if the wording was a bit different this question would have been solvable much easier than now, and this is why I think that:

it says "If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?"
considering the question like this most students would connect "divisor of a" to be the denominator that is b in the equation. So what happens we solve for b and say be must be 50 and a must be 143 and that is the wrong answer choice E. But the intention of the question is not to ask what is b, it says which of the following must be a divisor of a, it did NOT ask what must be the denominator in the equation. So what I'm tryinbg to say there is a bit of ambiguity, it is like double face, but how would we know if they ask for b or only for a factor of a. for the average even above average student I think this trap exists.
so if the queastion asked what must be a factor of a, which in fact factor is a divisor, but I as many others did connect divisor with b and looking for a solution for b and that was wrong.
So if the wording said what must be a factor of a than the things flow in other direction which is more digestable for the solutions given above.
Here it is again

$$\frac{a}{b}=2.86$$ or $$\frac{a}{b}=\frac{286}{100}=\frac{143}{50}$$

so we need the lowest term fraction cross multiply we get$$b=\frac{50a}{143}=\frac{50a}{11*13}$$

now teh fun part, in denominatr we have 11 and 13 , in nominator 50 and a. so 50 is not divisable with 11 nor 13 so that tells us that a MUST consist of 11 and 13. so in teh answer choices we can not have listed both together that would make two correct options. 13 is given, so it is the one factor that must be in a.

Let me know what other students think on this also experts what you think on this issue.

just my 2 cents
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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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26 Jun 2016, 01:05
a/b = 286/100 = 11*13*2/100=11*13/50. Since neither 11 nor 13 is a factor of 50 and since both 11 and 13 are prime, a is divisible by 11 and 13.

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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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11 Mar 2017, 12:36
a/b=2,86 or a/b=286/100

let's try answer choices: not A, E straightaway, B and D are multiplies (however 286/100=143/50 then D is out) 143/18=no integer
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Re: If a and b are positive integers such that a/b = 2.86, which  [#permalink]

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19 May 2017, 08:25
vigneshpandi wrote:
If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?

A. 10
B. 13
C. 18
D. 26
E. 50

$$\frac{a}{b} = \frac{286}{100}$$

Or, $$\frac{a}{b} = \frac{143}{50}$$

Now, $$a = 143 = 13*11$$

So, The divisor of a must be (B) 13
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Re: If a and b are positive integers such that a/b=2.86, which  [#permalink]

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06 Jun 2018, 09:07
a/b = 2.86 => 286/100

286/100 => 143/50. So a = 143*k where k is an integer. And 143 is 13*11. So it is divisible by 13
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If a and b are positive integers such that a/b=2.86, which  [#permalink]

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10 Jan 2020, 00:42
Hi Bunuel,
as per your post, "Divisors & Remainders", which was explained by

KARISHMA, VERITAS PREP ---- Knocking Off the Remaining Remainders

which was an outstanding post, eased me to find remainders.
Coming to our question, i have started solving the problem with the same appraoch,

a/b = 2.86
where 2 = quotient
remainder is expressed as .86, if we convert into simplest fraction we will be getting remainder and the divisor.
.86 = 86/100 = 43/50
where 43 = remainder & 50 = divisor
so our aim to find out the divisor of a,

here we have got divisor as 50, so as per the concept, any other divisor values will be a multiple of 50.
so from our option choices, only 50 is a required multiple. so i chose E as my answer choice. (by the way its wrong)

from the comment section i could see, 2.86 is simplified as 286/100 = 143/50, where in my approach i am doing it as 86/100 = 43/50. May i know whether my apprach is right or not.

Bunuel wrote:
vigneshpandi wrote:
If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?

1. 10
2. 13
3. 18
4. 26
5. 50

$$\frac{a}{b}=2.86=\frac{286}{100}=\frac{143}{50}$$ --> $$b=\frac{50a}{143}=\frac{50a}{11*13}$$, for $$b$$ to be an integer $$a$$ must have all the factors of 143 (50 has none of them). Hence $$a$$ must be divisible by both 11 and 13.

If a and b are positive integers such that a/b=2.86, which   [#permalink] 10 Jan 2020, 00:42

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