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# If a and b are positive integers such that a/b = 82.024, which of the

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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.

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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 $$\frac{6}{250}$$.

**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]
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.

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