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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550


Remainder= > R/B=24/1000=3/125 -> R*125=3B -> B must be a multiple of 125 Answer (D) 250
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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a/b = 82.024 -> a/b = 82,024/1000
We just have to simplify this equation to get a/b = 20,506/250.
Hence response D
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550


\(\frac{a}{b}\) = 82.024 => \(\frac{a}{b}\) = 82 + \(\frac{24}{1000}\)

Therefore, a = 82*b + \(\frac{24*b}{1000}\) => b should be a factor of 1000 and \(\frac{24*b}{1000}\) should be integer.

Out of all the answer choices, 100, 200, and 250 are factors of 100. But, \(\frac{24*b}{1000}\) returns an integer value only when b = 250.

So, the answer is (D).
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If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550

If \(a = bQ + r\), then \(\frac{a}{b} = Q + \frac{r}{b}\)

(a = dividend, b = divisor, Q = quotient, r = remainder)

Change the remainder from decimal to fraction: .024 =\(\frac{24}{1000} =\frac{r}{b}\)

In order for \(a\) and \(b\) to be integers, \((\frac{r}{b} * b)\) must result in an integer.

The decimal .084 = \(\frac{84}{1000}\) * (some # \(b\)) must have a result with no decimals. You cannot add a remainder with decimals to the product a = bQ: \(a\) would not be an integer, which the prompt says it is.

We know**:

1) \(r\) is an integer;

2) \(b\) is an integer;

3) \((\frac{r}{b} * b)\) is an integer, so \(b\) must be a multiple of the denominator in \((\frac{r}{b})\) to stay consistent with: \(a = bQ + r\)

4) No answer choice is b = 1,000 (which would make life easy)

If \(\frac{r}{b} = \frac{24}{1000}\), reduce in succession:

\(\frac{r}{b} =\frac{24}{1000} = \frac{12}{500} =\frac{6}{250} = \frac{3}{125}\)

Any answer choice that equals any one of those denominators, such that \(r\) becomes an integer, could equal \(b\).

Answer D, b = 250, as the only multiple of 125, makes \(r\) an integer.

OR, from the fraction reduced to simpler expressions in succession above, one expression is \(\frac{6}{250}\).

\(\frac{6}{250} *\\
250 = 6 = r\)

From here the dividend \(a\) can be "rebuilt."

b = 250, r = 6, Q is 82. \(a\) = (250)(82) + 6. \(a\) = 20,506. \(r\) is now an integer: \(b\) = 250 clears the fraction \(\frac{6}{250}\).

Answer D

**It is a little easier to see these concepts, especially the relationship between decimal values and integer values, with a simple example.

\(\frac{12}{5} = 2 +\\
R2\)
\(12 = (5)(2) + 2\)
\(a = bQ + r\)
\(\frac{a}{b} = Q + \frac{r}{b}\)
\(\frac{12}{5} = 2 + \frac{2}{5}\)
\(\frac{12}{5} = 2 + (.4)\)
\(\frac{12}{5} = 2.4\)
\(\frac{r}{b} =\frac{4}{10} = \frac{2}{5}\)

The denominator tells us that \(b\) here is a multiple of 5 -- \((\frac{2}{5})\) * (multiple of 5) = integer.
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
simply,
0.024*250=6 , integer
others results are not integer so...ans d
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550


\(\frac{a}{b} = 82.024\)=\(\frac{82024}{1000}=\frac{4(20506)}{4(250)}=\frac{20506}{2500}\)
Hence, b=250.

Ans. (D)
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If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
Expert Reply
Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550


Solution:

a/b = 82 + 0.024

a = 82b + 0.024b

Notice that 0.024b is the remainder when a is divided by b and it must be an integer. We see that b must be 250 since only 0.024 x 250 = 6 is an integer (note: all the other values of b will have 0.024b as a non-integer).

Alternate Solution:

Since a/b = 82.024, the quotient from the division of a by b is 82. Let R be the remainder from the division of a by b. Then, we have:

a = 82b + R

a/b = 82 + R/b

82.024 = 82 + R/b

R/b = 0.024 = 24/1000 = 3/125

Since R and b are integers, b must be a multiple of 125. The only multiple of 125 among the answer choices is 250.

Answer: D
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550


\(\frac{a}{b} = 82.024\)

\(\frac{a}{b} = 82 \frac{24}{1000}\)

\(\frac{a}{b} = 82 \frac{12}{500} = 82 \frac{6}{250} = 82 \frac{3}{125}\)

We can't reduce 3/125 any further; b must be a multiple of 125.

The only answer that is a multiple of 125 is D.
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
similar questions for this ?
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
If we assume remainder = x
Then x/b = 0.024
x/0.024 = b
=> x * 1000/24 = b
=> 250/6=b/x
So b = 250
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If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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Bunuel wrote:
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

(A) 100
(B) 150
(C) 200
(D) 250
(E) 550


I usually solve such questions as follows:

a/b=82.024
a/b=82+24/1000 {now in the form of dividend/divisor=quotient + remainder/divisor}
We can see that the divisor is 1000 but options have both 250 and 100 which can both become 1000, so we simplify further

a/b=82+6/250 (so 100 cannot be the value, therefore the answer is 250)

Is this approach correct? Bunuel
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
If a and b are positive integers such that a/b = 82.024, which of the following can be the value of b?

a/b = 82.024 = 82024*10^-3 = 10253*(2^3)*(10^-3)=10253*(5^-3)

Hence for a to be an integer, b must be = k(5^m) where k is a positive integer and m is a positive integer greater than or equal to 3

Now, we will analyze the options,
(A) 100= (2^2)(5^2)
(B) 150= (2)(3)(5^2)
(C) 200= (2^3)(5^2)
(D) 250= (2)(5^3)
(E) 550=(2)(11)(5^2)

Hence D
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Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]
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