Problem Solving

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 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.

j_shreyans wrote: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

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.

vinni.k wrote: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.

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

vinni.k wrote: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.

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.

vinni.k wrote: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.

vinni.k wrote: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.

vinni.k wrote: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.

Mitch and Matt,

Thanks. I got it. . Really appreciate it.

vinni.k wrote:Mitch and Matt,

Thanks. I got it. . Really appreciate it.

No problem! Great question.