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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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

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

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

Ans. (D)
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