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If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer?

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Bunuel
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If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   18 Sep 2018, 00:44
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If \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\), which of the following must be an integer?


I. \(\frac{x}{8}\)

II. \(\frac{x}{12}\)

III. \(\frac{x}{24}\)


A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III
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B
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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   18 Sep 2018, 01:05
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Bunuel wrote:
If \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\), which of the following must be an integer?


I. \(\frac{x}{8}\)

II. \(\frac{x}{12}\)

III. \(\frac{x}{24}\)


A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III


\(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\)
i.e. 13/12 = 13/x
i.e. x = 12

i.e. x will be divisible by ONLY 12 and every factor of 12 hence

Answer: option B
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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   18 Sep 2018, 02:15
Expert Reply

Solution


Given:
    • \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\)

To find:
    • Which of the given options is always an integer?

Approach and Working:
    • Multiplying the equation, \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\), by x on both sides gives,
      o \(\frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 13\)
      o \(\frac{13x}{12} = 13\)
      o Implies, x = 12

Thus, only II is true

Hence, the correct answer is option B.

Answer: B

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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   18 Sep 2018, 03:12
Bunuel wrote:
If \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\), which of the following must be an integer?


I. \(\frac{x}{8}\)

II. \(\frac{x}{12}\)

III. \(\frac{x}{24}\)


A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III


1/2+1/3+1/4=13/12. Thus x=12.

Not just put 12 in place of x in all three options we get only 2 II to be an integer.

Option B.
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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   20 Sep 2018, 01:13
If the question used "COULD be true" instead of "must be true", would the answer be D?

Because we know that \(\frac{13}{12}\) = \(\frac{13}{x}\)

So x could also be a multiple of 12, therefore \(\frac{x}{24}\) could also be true.

Am I correct?
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If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   20 Sep 2018, 22:50
1/2 + 1/3 + 1/4 = 13/x

=>13/12 = 13/x
=>13x = 156
=>x = 156/13
=>x=12

A. 12/8 = 3/2 - Not an Integer
B. 12/12 = 1 - Integer
C. 12/24 = 1/2 - Not an integer

Hence B is the answer
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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   23 Sep 2018, 16:32
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Bunuel wrote:
If \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\), which of the following must be an integer?


I. \(\frac{x}{8}\)

II. \(\frac{x}{12}\)

III. \(\frac{x}{24}\)


A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III


Multiplying by 12x we have:

6x + 4x + 3x = 156

13x = 156

x = 12; thus, x/12 must be an integer.

Answer: B
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If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   13 Sep 2019, 23:44
Bunuel wrote:
If \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\), which of the following must be an integer?


I. \(\frac{x}{8}\)

II. \(\frac{x}{12}\)

III. \(\frac{x}{24}\)


A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III


Given: \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\)

Asked: Which of the following must be an integer?

\(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{x}\)

\(\frac{(6+4+3)}{12} = \frac{13}{x}\)

\(\frac{13}{12} = \frac{13}{x}\)

\(\frac{1}{12} = \frac{1}{x}\)

x = 12

I. \(\frac{x}{8}\)

\(\frac{12}{8} = \frac{3}{2} = 1.5\)

NOT AN INTEGER

II. \(\frac{x}{12}\)

\(\frac{12}{12} = 1\)

INTEGER

III. \(\frac{x}{24}\)

\(\frac{12}{24} = \frac{1}{2} = 0.5\)

NOT AN INTEGER


IMO B
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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink] New post   14 May 2023, 09:18
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Re: If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? [#permalink]
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