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If x is a positive integer, what is the value of x ?

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Tough and Tricky questions: Decimals.



If \(x\) is a positive integer, what is the value of \(x\)?


(1) The first nonzero digit in the decimal expansion of \(\frac{1}{x!}\) is in the hundredths place.

(2) The first nonzero digit in the decimal expansion of \(\frac{1}{(x+1)!}\) is in the thousandths place.

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: If x is a positive integer, what is the value of x ? [#permalink]

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Statement 1:

The first nonzero digit in the decimal expansion of \(\frac{{1}}{{x!}}\) is in the hundredths place.

Let us quickly check few factorials.

\(2! =2\) ---->\(\frac{1}{2}\) =.5

\(3!= 6\) ----> \(\frac{1}{6}\)= .16

\(4! = 24\) -----> \(\frac{1}{24}\) = .04 -- first nonzero digit is hundredths place
\(5!= 120\) -----> \(\frac{1}{120}\) = .008 ---- Hundredths place is zero.

With increasing value of denominator, place of nonzero digit in decimal expansion will shift to right.

So only \(4!\)has first non zero digit in hundredths place.

Therefore, \(x= 4\)

Statement1 is sufficient.



Statement 2:
The first nonzero digit in the decimal expansion of \(\frac{{1}}{{(x+1)!}}\) is in the thousandths place.

Let us try some number and check.

\(x= 4\) ----->\(\frac{1}{(4+1)!}\) \(=\) \(\frac{1}{5!}\) \(=\) \(0.008\) after decimal expansion first non zero digit is at thousandths place.

\(x= 5\) -----> \(\frac{1}{(5+1)!}\) = \(\frac{1}{6!}\) \(=\) \(0.0013\) after decimal expansion first non zero digit is at thousandths place.

Statement 2 not sufficient.

Answer: A

Regards,
Ammu

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Bunuel wrote:

Tough and Tricky questions: Decimals.



If \(x\) is a positive integer, what is the value of \(x\)?

(1) The first nonzero digit in the decimal expansion of \(\frac{1}{x!}\) is in the hundredths place.

(2) The first nonzero digit in the decimal expansion of \(\frac{1}{(x+1)!}\) is in the thousandths place.

Kudos for a correct solution.


Official Solution:

If \(x\) is a positive integer, what is the value of \(x\)?

The question does not need rephrasing, although we should note that \(x\) is a positive integer.

Statement 1: SUFFICIENT. We should work from the inside out by first listing the first several values of \(x!\) (the factorial of \(x\), defined as the product of all the positive integers up to and including \(x\)).
\(1! = 1\)
\(2! = 2\)
\(3! = 6\)
\(4! = 24\)
\(5! = 120\)
\(6! = 720\)
\(7! = 5040\)

Now we consider decimal expansions whose first nonzero digit is in the hundredths place. Such decimals must be smaller than \(0.1\) (\(\frac{1}{10}\)) but at least as large as \(0.01\) (\(\frac{1}{100}\)). Therefore, for \(\frac{1}{x!}\) to lie in this range, \(x!\) must be larger than 10 but no larger than 100. The only factorial that falls between 10 and 100 is \(4! = 24\), so \(x = 4\).

(Note that factorials are akin to exponents in the order of operations, so \(\frac{1}{x!}\) indicates "1 divided by the factorial of \(x\)," not "the factorial of \(\frac{1}{x}\)," which would only have meaning if \(\frac{1}{x}\) were a positive integer.)

Statement 2: INSUFFICIENT. We consider decimal expansions whose first nonzero digit is in the thousandths place. Such decimals must be smaller than \(0.01\) (\(\frac{1}{100}\)) but at least as large as \(0.001\) (\(\frac{1}{1000}\)). Therefore, for \(\frac{1}{(x+1)!}\) to lie in this range, \((x+1)!\) must be larger than 100 but no larger than 1,000.

There are two factorials that fall between 100 and 1,000, namely \(5! = 120\) and \(6! = 720\). Thus, \(x+1\) could be either 5 or 6, and \(x\) could be either 4 or 5.

Answer: A.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If x is a positive integer, what is the value of x ?   [#permalink] 26 Sep 2017, 23:17
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