Positive integers

[j_shreyans]

Guys ,

If a & b are the positive integers such that a/b=2.86 which of the following must be divisor of a.

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

[GMATGuruNY]

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

a/b = 2.86 = 286/100 = 143/50.
Since 143/50 cannot be reduced further, a must be a multiple of 143, while b must be a multiple of 50.
Since a is a multiple of 143 -- and 143 = 11*13 -- answer choice B must divide evenly into a.

The correct answer is B.

[Brent@GMATPrepNow]

If a & b are the positive integers such that a/b = 2.86 which of the following must be divisor of a.

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

Given: a/b = 2.86
Rewrite as fraction: a/b = 286/100
Simplify: a/b = 143/50
Cross multiply to get: 50a = 143b

This tells us that 50a is a multiple of 143, which is the same as saying that 50a is divisible by 143.
Since 50 is not divisible by 143, a must be divisible by 143
Since 143 = (11)(13), a must be divisible by 11 and by 13

Check the answer choices . . . the correct answer is B

Cheers,
Brent

[GMATinsight]

Guys ,

If a & b are the positive integers such that a/b=2.86 which of the following must be divisor of a.

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

If a/b = 2.86
then, a / 2.86 = b
i.e. a must be divisible by 2.86 = 286/100 = 143/50
286 = is multiple of 13 therefore only options B and D are possible options but since 143 is odd multiple of 13 therefore only option B is possible.

Answer: Option B

[vinni.k]

One question. Why can't answer be E ? Is it because it is not asking about what is a remainder.

What is wrong here ?

Given: a/b = 2.86
Now, R/B = 86/100 = 43/50
Cross-multiply, R = 43/50 * B
Now, i know i will get remainder, but it still gives me the divisor as 50

Just want to know what is wrong here.

[GMATGuruNY]

One question. Why can't answer be E ? Is it because it is not asking about what is a remainder.

What is wrong here ?

Given: a/b = 2.86
Now, R/B = 86/100 = 43/50
Cross-multiply, R = 43/50 * B
Now, i know i will get remainder, but it still gives me the divisor as 50

Just want to know what is wrong here.

The question ask which of the answer choices must be a DIVISOR OF A.
To answer this question, we must determine the LEAST POSSIBLE VALUE OF A.

You have correctly determined that R = (43/50)B.
Since R must be a positive integer, the least possible option for B is 50, with the result that R = (43/50)(50) = 43.

Plugging B = 50 into A/B = 2.86, we get:
A/50 = 2.86
A = (2.86)(50) = (286/100)(50) = 286/2 = 143.

Of the 5 answer choices, only B must divide into 143.

[vinni.k]

Mitch,

Thanks for the explanation. I got your point. But what is the difference between the following question and the question that is posted above. If i go according to this following question, then answer is 50 (E)

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the reminder when A is divided by B?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

OA is B

[GMATGuruNY]

Mitch,

Thanks for the explanation. I got your point. But what is the difference between the following question and the question that is posted above. If i go according to this following question, then answer is 50 (E)

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the reminder when A is divided by B?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

OA is B

The question posted at top asks for a DIVISOR OF A.
The question posted here asks for a possible REMAINDER when A is divided by B.
The two questions ask for completely different values.

For the problem posted here, use the following formula:
Remainder/Divisor = Decimal
(For the reasoning behind this formula, check here: https://www.beatthegmat.com/quotient-rem ... 01371.html.)

Plugging R = remainder, B = divisor, and Decimal = 35/100 = 7/20 into the formula in blue, we get:
R/B = 7/20.
Implication:
R must be a multiple of 7, while B must be a multiple of 20.

Of the five answer choices, only B is a multiple of 7.

The correct answer is B.

[Jay@ManhattanReview]

Mitch,

Thanks for the explanation. I got your point. But what is the difference between the following question and the question that is posted above. If i go according to this following question, then answer is 50 (E)

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the reminder when A is divided by B?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

OA is B

4.35 = 4 + 0.35 = 4 + (35/100).

If A = 435 and B = 100, the remainder = 35; however, if A = 870 and B = 200, the remainder = 70.

The reduced form of 35/100 = 7/20. We see that 7, 35, and 70 are multiples of '7', thus the possible remainder could 7, 14, ...

When we write 7/20 as 14/40, we get 14 as the remainder.

The correct answer: B

Relevant book: Manhattan Review GMAT Number Properties Guide

Hope this helps!

-Jay
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[vinni.k]

Hi Mitch,

Yes, the two questions are different, but my point of showing that question was the way to get the divisor.
In the original question it is given A/B = 2.86 and we are asked which of the following must be divisor of "A". In other words, what is the value of "B", which is a divisor.

What i am asking you is that - why we need to plug back 50 in B when we have already got the divisor ?

Algebraically, the relationship can be written as

x/N = Q + R/N

x - dividend
N - divisor
Q - Quotient
R - Remainder

Given a/b = 2.86 = 2 (quotient) + 86/100 (Remainder/Divisor)
Now, 86/100 = 43/50

Just want to clarify my doubt.

[GMATGuruNY]

Hi Mitch,

Yes, the two questions are different, but my point of showing that question was the way to get the divisor.
In the original question it is given A/B = 2.86 and we are asked which of the following must be divisor of "A". In other words, what is the value of "B", which is a divisor.

What i am asking you is that - why we need to plug back 50 in B when we have already got the divisor ?

Algebraically, the relationship can be written as

x/N = Q + R/N

x - dividend
N - divisor
Q - Quotient
R - Remainder

Given a/b = 2.86 = 2 (quotient) + 86/100 (Remainder/Divisor)
Now, 86/100 = 43/50

Just want to clarify my doubt.

Your confusion is understandable.
The term divisor has two meanings:

1. A number by which another number is to be divided.
Example: x/y
Here -- since x is to be divided by y -- y is known as the DIVISOR.

2. An integer that divides EVENLY into another integer, with no remainder.
Example: 21/7 = 3
Here -- since 7 divides evenly into 21 -- 7 is considered a DIVISOR OF 21.
Note:
A divisor of x is also known as a FACTOR OF X.

You have determined the following:
a/b = 2 + (86/100)
a = 2b + (43/50)b.

In the second equation, the value in red is a divisor, in accordance with definition #1.

However, the question stem asks for a SPECIFIC TYPE of divisor: a divisor of a.
In this case, we are dealing with definition #2.
A divisor of a refers to a FACTOR of a: an integer that divides EVENLY into a, with no resulting remainder.
The value in red is not necessarily a factor of a.

To determine what values must divide evenly into a, we must calculate the smallest possible value of a.
In the second equation, the least possible value for b that will yield an integer value for a is 50.
Substituting b=50 into the second equation, we get:
a = 2*50 + (43/50)(50) = 143.

Of the five answer choices, only A must divide evenly into 143.

[Matt@VeritasPrep]

Mitch,

Thanks for the explanation. I got your point. But what is the difference between the following question and the question that is posted above. If i go according to this following question, then answer is 50 (E)

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the reminder when A is divided by B?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

OA is B

It might be easier to see if we use a ratio instead:

A/B = 4.35

A = 4.35B

100A = 435B

20A = 87B

So our smallest positive solution is A = 87, B = 20, which gives remainder 7. From there, we can make the remainder any multiple of 7 we please: for instance, another solution is A = 87*2, B = 20*2, which gives A/B = 174/40, which has remainder 14.

[Matt@VeritasPrep]

Hi Mitch,

Yes, the two questions are different, but my point of showing that question was the way to get the divisor.
In the original question it is given A/B = 2.86 and we are asked which of the following must be divisor of "A". In other words, what is the value of "B", which is a divisor.

and just to reiterate my response to your first post upthread:

A = 2.86B

100A = 286B

50A = 143B

which gives the smallest positive solution A = 143, B = 50. That means any value of A will be a multiple of 143, which itself is a multiple of 11 and 13. So 11 and 13 MUST be divisors of any valid A.

Seems a lot easier to me to do it this way.

[vinni.k]

Mitch and Matt,

Thanks. I got it. . Really appreciate it.

[Matt@VeritasPrep]

Mitch and Matt,

Thanks. I got it. . Really appreciate it.

No problem! Great question.