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We are dealing with a repeating decimal in this question. It's helpful to know that there's a way to write these kinds of decimals as a fraction. For example, the repeating decimal 0.444444444(4) may be written as \(\frac{4}{9}\). So, \(\frac{5}{9}\), \(\frac{7}{9}\) and \(\frac{8}{9}\) will all be repeating decimals. You might check it in your calculator. In order to make two decimal points repeat, you have to divide the two digit number by 99. For example, \(\frac{23}{99} = 0.232323232323(23)\). Similarly, \(\frac{36}{99} = \frac{4}{11} = 0.36363636(36)\). Now it's clear that the minimum value of \(m = 4\).
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In case you did not know the formula for the repeating decimal (most probably did not), there is another approach to solving this question - backsolving. This is not a typical backsolving question, however, since both of the variables are unknown and we have to make some assumptions to get to the solution.

1. Looking at the repeating decimal - 0.36.... - the ratio between m and n has to be slightly less than 1:3. 2. Let's run through the answer choices: A. 3 - the number that's slightly less than 3*3 is 8. \(\frac{3}{8} = 0.375\). Does not work B. 4 - the number that's slightly less than 3*4 is 11. \(\frac{4}{11} = 0.3636\). Works! C. 7 D. 13 E. 22

We could continue going through answer choices C, D, and E, but the question asks us for the smallest possible value of m, and 4 is the smallest of the ones that work (even if multiple do) so there is no value to check others.

GMAT Diagnostic Test Question 8 Field: number properties Difficulty: 700

Rating:

If \(m\) and \(n\) are integers, what is the smallest possible value of integer \(m\) if \(\frac{m}{n}\) = 0.3636363636...?

A. 3 B. 4 C. 7 D. 13 E. 22

In the original question (which I downloaded from download/file.php?id=9039) , it doesn't say that n is an integer... that mean that m and n could be 3 and 33/4 respectivelly. In that case A would be the correct choice.

In the original question (which I downloaded from download/file.php?id=9039) , it doesn't say that n is an integer... that mean that m and n could be 3 and 33/4 respectivelly. In that case A would be the correct choice.

hi, my solution is backsolving with picking numbers:

First of all, since the division is <|1|, then |m|<|n|

I started with 3 being divided by many numbers: 3/4 = 0,7xxx (not the answer) 3/5=0,6xxx (uh oh) 3/7 =0,4xxx (nope)

Do you see a pattern here? It is kind of logical, but just to be sure you noticed that: the greater the dividend, the smaller is the result of the division. Going on:

3/8 =0,37x (almost!) 3/9 = 3 we "crossed the line", move on to the next alternative

(x means that I stopped the division, even though I knew I could continue dividing)

since 4/8 = 0.5, let's start from here

4/9 = 0,4x (no) 4/10 = 0,4 (no) 4/11 = 0,3636363636363636363636363636363636363636363636 (ok, I did not the division until shown above :D)

so 4 is the answer.

It took some time... number properties always takes more than 2 minutes each for me... I try to overcome doing faster the others...

hi, my solution is backsolving with picking numbers:

First of all, since the division is <|1|, then |m|<|n|

I started with 3 being divided by many numbers: 3/4 = 0,7xxx (not the answer) 3/5=0,6xxx (uh oh) 3/7 =0,4xxx (nope)

Do you see a pattern here? It is kind of logical, but just to be sure you noticed that: the greater the dividend, the smaller is the result of the division. Going on:

3/8 =0,37x (almost!) 3/9 = 3 we "crossed the line", move on to the next alternative

(x means that I stopped the division, even though I knew I could continue dividing)

since 4/8 = 0.5, let's start from here

4/9 = 0,4x (no) 4/10 = 0,4 (no) 4/11 = 0,3636363636363636363636363636363636363636363636 (ok, I did not the division until shown above :D)

so 4 is the answer.

It took some time... number properties always takes more than 2 minutes each for me... I try to overcome doing faster the others...

Thanks for sharing your process. I just looked at fraction/decimal/% equivalents last night and was pissed that I didn't remember this one! Nonetheless, I appreciate the lesson learned about repeating decimals. I'm definitely adding it to my list of hints/tricks.
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We are dealing with a repeating decimal in this question. It's helpful to know that there's a way to write these kinds of decimals as a fraction. For example, the repeating decimal 0.444444444(4) may be written as \(\frac{4}{9}\). So, \(\frac{5}{9}\), \(\frac{7}{9}\) and \(\frac{8}{9}\) will all be repeating decimals. You might check it in your calculator. In order to make two decimal points repeat, you have to divide the two digit number by 99. For example, \(\frac{23}{99} = 0.232323232323(23)\). Similarly, \(\frac{36}{99} = \frac{4}{11} = 0.36363636(36)\). Now it's clear that the minimum value of \(m = 4\).

is this a rule or something that any divisor when divided by 99 or a factor of it (9 or 11) will end up giving a repeating decimal? Was this rule taught in high school...I don't remember. Where can we brush up these special rules. Any book to practice from?

I was taught this rule in a dedicated math class, not a regular high school class. I'm not sure if there is a book dedicated to special rules like this one. You're welcome to take a look at this thread listing the best GMAT Math books: best-gmat-math-prep-books-reviews-recommendations-77291.html

Good luck with your preparation!
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