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

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Math Expert
Joined: 02 Sep 2009
Posts: 46264
If a and b are positive integers such that a/b = 82.024, which of the [#permalink]

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20 Oct 2015, 12:33
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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

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Joined: 12 Sep 2015
Posts: 2567
Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]

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20 Oct 2015, 12:49
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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

Notice that we can rewrite 82.024 as 82 + 24/1000
So, 82.024 = 82 + 24/1000
Simplify to get: 82.024 = 82 + 3/125
Rewritten as an ENTIRE fraction, we get: 82.024 = [(82)(125) + 3]/125

a/b = [(82)(125) + 3]/125, so b COULD equal 125.
When we check the answer choices, we don't see 125.
That's okay, because we can rewrite [(82)(125) + 3]/125 as an EQUIVALENT fraction, just like we can rewrite 3/7 as 6/14 or 9/21 or 12/28 etc.

Likewise, if we rewrite [(82)(125) + 3]/125 as an EQUIVALENT fraction, the denominator (b) can be ANY MULTIPLE of 125
Checking the answer choices, only 250 is a multiple of 125

Cheers,
Brent
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Joined: 11 Sep 2013
Posts: 110
Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]

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20 Oct 2015, 12:55
1
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

b must be the factor of 1000 because 82.024 = 82,024/1000, a must be integer
=> I crossed out: B, E

If b = 100 => a=8202.4: not integer: OUT
If b = 200 => a is not integer becayse 82,024 is not divisible by 5
If b = 250 => a=20506: integer: OK

Ans: 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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20 Oct 2015, 13:31
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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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Intern
Joined: 12 Jun 2016
Posts: 2
Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]

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26 Jun 2016, 08:12
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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Joined: 05 May 2016
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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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05 Jul 2016, 12:35
1
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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Joined: 22 May 2016
Posts: 1757
If a and b are positive integers such that a/b = 82.024, which of the [#permalink]

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01 Oct 2017, 07:27
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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Intern
Joined: 20 Aug 2017
Posts: 2
Re: If a and b are positive integers such that a/b = 82.024, which of the [#permalink]

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03 Oct 2017, 07:16
simply,
0.024*250=6 , integer
others results are not integer so...ans d
Re: If a and b are positive integers such that a/b = 82.024, which of the   [#permalink] 03 Oct 2017, 07:16
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