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If x is an integer and 0 < x ≤ 100, what is the probability that x(x +

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If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 25 Dec 2017, 00:41
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 25 Dec 2017, 02:28
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


When x is a multiple of 4 for eg. 4,8,12.... then there are 25 such numbers between 0<x≤100

when x+1 is a multiple of 4 i.e x=4k-1 (where k is some constant) for eg. 3,7,11....., then again there will be 25 such numbers between 0<x≤100

So total numbers of the form x(x+1) that are multiple of 4 =25+25=50

So probability \(= \frac{50}{100}=\frac{1}{2}\)

Option D
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 25 Dec 2017, 05:12
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Solution



Given:

    • The variable x is greater than 0, but less than or equal to 100.
      o The values of x can be 1,2,3,…..so on till 100.
      o Thus we can say that the total possible values of x are 100.

Find:

    • We are given the expression x (x+1) and we need to find the probability that the expression x(x+1) is a multiple of 4.

Approach:

    • Since x and (x+1) are consecutive numbers, both cannot be even or a multiple of 4.
    • For x to be a multiple of 4, either x or x+1 has to be a multiple of 4.
    • Thus, let us look at the two possible cases:
      o x is a multiple of 4.
         x = 4k and k can be 1,2,3,4….till 25
         Since x = 4 * 25 = 100, thus the last value of k will be 25
         And total possible values can be 25.
      o (x + 1) is a multiple of 4.
         x + 1 = 4q
         x = 4q -1
         Thus, the values of q can be 1,2,3 and so on till…25 and the value of x will be 3,7…99
         Even in this case, the total possible values are 25.
    • Thus, total favourable cases = 25 + 25 = 50
    • And the probability that x(x+1) = Favorable cases/ Total case = 50/100 = 1/2

The correct answer is Option D.
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 25 Dec 2017, 15:57
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


From 1 to 100 there are: \(Number.Terms=Last.term-First.term+1=100-1+1=100\) total outcomes.

If x is a multiple of 4 then x(x+1) = multiple of 4.
If x+1 is a multiple of 4 then x(x+1) = multiple of 4.

From 1 to 100 there are: \(Number.Multiples=\frac{Largest.multiple-Smallest.multiple}{Multiple}+1=\frac{100-4}{4}+1=25\) multiples of 4 that fit \(x\), and \(25\) multiples of 4 that fit \(x-1\).

Probability is: \(\frac{Favorable.Outcomes}{Total.Outcomes}=\frac{(25+25)}{100}=1/2\).

(D) is the answer.
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post Updated on: 26 Dec 2017, 09:17
1
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Hi EgmatQuantExpert, niks18, chetan2u,

Can you please help me to clarify my doubt.

The question states that \(0 < x ≤ 100\), so for all the Integers value of \(x\) the answer will be \(\frac{50}{100}\). However, here x can be decimals also for ex - \(x\)= \((\sqrt{17}-1)/2\), x = \((\sqrt{65}-1)/2\).

Can you please let me know if I am missing anything here.
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Originally posted by rahul16singh28 on 26 Dec 2017, 08:39.
Last edited by rahul16singh28 on 26 Dec 2017, 09:17, edited 2 times in total.
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 08:46
rahul16singh28 wrote:
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Hi EgmatQuantExpert, niks18, chetan2u,

Can you please help me to clarify my doubt.

The question states that \(0 < x ≤ 100\), so for all the Integers value of \(x\) the answer will be \(\frac{50}{100}\). However, here x can be decimals also for ex - \(x\)= \(\frac{\sqrt{17[}{square_root]-1/2}\), x = \([square_root]65}-1/2\).

Can you please let me know if I am missing anything here.


Hi rahul16singh28

Here x is a multiple of 4. Multiples of 4 cannot be decimals (0.4=2/5 is not a multiple of 4), they will have to be an integer. so probability has to be between favorable integer values which are (25+25) and total integers possible which is 100
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 08:47
X = 4,8,....100 - 25 count
X+1 = 4,8,.....100- 25 count

Total = 25 + 25 = 50

Answer : 50/100 = 1/2
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 08:52
niks18 wrote:
rahul16singh28 wrote:
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Hi EgmatQuantExpert, niks18, chetan2u,

Can you please help me to clarify my doubt.

The question states that \(0 < x ≤ 100\), so for all the Integers value of \(x\) the answer will be \(\frac{50}{100}\). However, here x can be decimals also for ex - \(x\)= \(\frac{\sqrt{17[}{square_root]-1/2}\), x = \([square_root]65}-1/2\).

Can you please let me know if I am missing anything here.


Hi rahul16singh28

Here x is a multiple of 4. Multiples of 4 cannot be decimals (0.4=2/5 is not a multiple of 4), they will have to be an integer. so probability has to be between favorable integer values which are (25+25) and total integers possible which is 100


Hi niks18,

I think here we have to find out if x(x+1) is a multiple of 4. If we take \(x\)= \(\sqrt{17}-1/2\), then x(x+1) is a multiple of 4.
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 09:02
Quote:
Hi niks18,

I think here we have to find out if x(x+1) is a multiple of 4. If we take \(x\)= \(\sqrt{17}-1/2\), then x(x+1) is a multiple of 4.


Hi rahul16singh28

if \(x= \sqrt{17}-\frac{1}{2}\), then \(x+1=\sqrt{17}-\frac{1}{2}+1=\sqrt{17}+\frac{1}{2}\)

now \(x(x+1)=(\sqrt{17}-\frac{1}{2})(\sqrt{17}+\frac{1}{2})=17-\frac{1}{4}\) which is not a multiple of \(4\).

Point is non integers cannot be multiples of integers otherwise every non integer can be multiple of integer
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 09:15
niks18 wrote:
Quote:
Hi niks18,

I think here we have to find out if x(x+1) is a multiple of 4. If we take \(x\)= \(\sqrt{17}-1/2\), then x(x+1) is a multiple of 4.


Hi rahul16singh28

if \(x= \sqrt{17}-\frac{1}{2}\), then \(x+1=\sqrt{17}-\frac{1}{2}+1=\sqrt{17}+\frac{1}{2}\)

now \(x(x+1)=(\sqrt{17}-\frac{1}{2})(\sqrt{17}+\frac{1}{2})=17-\frac{1}{4}\) which is not a multiple of \(4\).

Point is non integers cannot be multiples of integers otherwise every non integer can be multiple of integer


Hi niks18,

May be I am missing something silly here but what I meant was if \(x\)= \((\sqrt{17} - 1)/2\). True that non-integers cannot be multiple of Integer but here we have a case where product of 2 non-integer (x & x+1) is a multiple of integer for \(x\)= \((\sqrt{17} - 1)/2\) and this is what exactly we need to find.
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 09:18
2
rahul16singh28 wrote:
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Hi EgmatQuantExpert, niks18, chetan2u,

Can you please help me to clarify my doubt.

The question states that \(0 < x ≤ 100\), so for all the Integers value of \(x\) the answer will be \(\frac{50}{100}\). However, here x can be decimals also for ex - \(x\)= \((\sqrt{17}-1)/2\), x = \((\sqrt{65}-1)/2\).

Can you please let me know if I am missing anything here.


It was missing in the question that x is an integer.
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 26 Dec 2017, 09:20
2
Bunuel wrote:
rahul16singh28 wrote:
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Hi EgmatQuantExpert, niks18, chetan2u,

Can you please help me to clarify my doubt.

The question states that \(0 < x ≤ 100\), so for all the Integers value of \(x\) the answer will be \(\frac{50}{100}\). However, here x can be decimals also for ex - \(x\)= \((\sqrt{17}-1)/2\), x = \((\sqrt{65}-1)/2\).

Can you please let me know if I am missing anything here.


It was missing in the question that x is an integer.

:lol: :lol: :lol:

rahul16singh28 our problem is solved :thumbup: :thumbup: :thumbup: :thumbup:
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +  [#permalink]

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New post 29 Dec 2017, 22:33
rahul16singh28 wrote:
Bunuel wrote:
If 0 < x ≤ 100, what is the probability that x(x + 1) is a multiple of 4?

A. 1/8
B. 1/4
C. 33/100
D. 1/2
E. 29/50


Hi EgmatQuantExpert, niks18, chetan2u,

Can you please help me to clarify my doubt.

The question states that \(0 < x ≤ 100\), so for all the Integers value of \(x\) the answer will be \(\frac{50}{100}\). However, here x can be decimals also for ex - \(x\)= \((\sqrt{17}-1)/2\), x = \((\sqrt{65}-1)/2\).

Can you please let me know if I am missing anything here.



I solved this question with the assumption that x is an integer. :)

You did the right thing by not assuming it in the starting and considering all the possible cases. But given that GMAT does not test us on such complex matters, I felt that assuming x is an integer makes sense and it might just be miss a while framing the question, which Bunuel did clarify. :)


Regards,
Saquib
e-GMAT Quant Expert
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Re: If x is an integer and 0 < x ≤ 100, what is the probability that x(x +   [#permalink] 29 Dec 2017, 22:33
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