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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   95% (hard)

Question Stats: 36% (03:22) correct 64% (03:04) wrong based on 36 sessions

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If $$a$$ and $$b$$ are single-digit positive numbers and $$\frac{a}{b}$$ is NOT a recurring decimal, what is the value of $$a$$?

(1) $$-\frac{1}{3} > -\frac{a}{b} > -\frac{4}{5}$$

(2) $$b$$ is equal to the sum of its positive divisors excluding $$b$$ itself.

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Math Expert V
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Official Solution:

If $$a$$ and $$b$$ are single-digit positive numbers and $$\frac{a}{b}$$ is NOT a recurring decimal, what is the value of $$a$$?

$$a$$ and $$b$$ are single-digit positive numbers means that $$a$$ and $$b$$ can be 1, 2, 3, 4, 5, 6, 7, 8, or 9.

(1) $$-\frac{1}{3} > -\frac{a}{b} > -\frac{4}{5}$$

Simplify by multiplying by -1: $$\frac{1}{3} < \frac{a}{b} < \frac{4}{5}$$

Convert to decimals $$0.(3) < \frac{a}{b} < 0.8$$.

Since $$\frac{a}{b}$$ is NOT a recurring decimal, then it can be 0.4 ($$a = 2$$, $$b = 5$$), 0.5 ($$a = 1$$, $$b = 2$$), ... Not sufficient.

(2) $$b$$ is equal to the sum of its positive divisors excluding $$b$$ itself.

From single digit numbers only 6 satisfies this condition: $$6 = 1 + 2 + 3$$. Since $$\frac{a}{b}$$ is NOT a recurring decimal, then $$\frac{a}{6}$$ can be $$\frac{3}{6} = 0.5$$, $$\frac{6}{6} = 1$$, or $$\frac{9}{6} = 1.5$$. Not sufficient.

(1)+(2) From (2) $$\frac{a}{b}$$ can be $$\frac{3}{6} = 0.5$$, $$\frac{6}{6} = 1$$, or $$\frac{9}{6} = 1.5$$. Only one of which is between 0.(3) and 0.8, namely 0.5. Sufficient.

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From single digit numbers only 6 satisfies this condition: 6=1+2+3.

However, 3 is also valid: 3 = 1+2.

What am I missing here? Thanks in advance Math Expert V
Joined: 02 Sep 2009
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z0rpia wrote:

From single digit numbers only 6 satisfies this condition: 6=1+2+3.

However, 3 is also valid: 3 = 1+2.

What am I missing here? Thanks in advance (2) says: b is equal to the sum of its positive divisors excluding b itself.

2 is not a divisor of 3. So, 3 does NOT equal to the sum of its positive divisors excluding b itself. 3 has only one positive divisor excluding 3 itself namely 1.
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Bunuel wrote:

Since $$\frac{a}{b}$$ is NOT a recurring decimal, then it can be 0.4 ($$a = 2$$, $$b = 5$$), 0.5 ($$a = 1$$, $$b = 2$$), ... Not sufficient.

Does 3/5 & 7/10 satisfy fact 1? I think so as they are 0.6 & 0.7 respectively. Did not you consider them fro certain reason?
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Mo2men wrote:
Bunuel wrote:

Since $$\frac{a}{b}$$ is NOT a recurring decimal, then it can be 0.4 ($$a = 2$$, $$b = 5$$), 0.5 ($$a = 1$$, $$b = 2$$), [highlight]... Not sufficient.[/highlight]

Does 3/5 & 7/10 satisfy fact 1? I think so as they are 0.6 & 0.7 respectively. Did not you consider them fro certain reason?

a/b can be 3/5. You can see ... part in the solution above, which indicates that there are some other values of a/b possible.

a/b cannot be 7/10, because 10 is not a single digit number.
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A quick tip -

To make sure of statement 2 faster. You can use terminating decimals theory. Because recurring decimals are basically non-terminating. So check if the denominator of any number is having multiples of 2 or 5 after u reduce fraction to lowest form.

Example-
We can simply eliminate 1/6 , 2/6 , 4/6 , 5/6, 7/6 and 8/6 because they have no denominator divisible by 2 and 5.
4/6 when reduced gives 2/3.
8/6 gives 4/3. and so on.

Hey Bunuel
To check statement 2 and make sure that we do have a recurring decimal. We can use terminating decimals theory right?
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Bunuel,
Pls explain "a/b is not a recurring decimal"
What are the chararictics of NOT RECURRING decimal.

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Math Expert V
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BelalHossain046 wrote:
Bunuel,
Pls explain "a/b is not a recurring decimal"
What are the chararictics of NOT RECURRING decimal.

Posted from my mobile device

A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. For example, 1/3 = 0.333333....., 2/9 = 0.2222.......

If it's not the case then the decimal is terminating: 0.7, 0.139, 0.999, 0.55, ...

Theory:
Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$. Other examples of terminating decimals: fractions $$\frac{1}{8}$$, $$\frac{1}{25}$$, ...

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be the terminating decimal.

Check below for more:
Math: Number Theory
How to Identify Terminating Decimals on the GMAT
Terminating Decimals in Data Sufficiency on the GMAT
Terminating and Recurring Decimals

Even more:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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