If x is an integer and 0

Question

If x is an integer and 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

niks18 · Answer

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

EgmatQuantExpert · Answer

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   .

exc4libur · Answer

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.

rahul16singh28 · Answer

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.

niks18 · Answer

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

ashishpanakanti · Answer

X = 4,8,....100 - 25 count
X+1 = 4,8,.....100- 25 count

Total = 25 + 25 = 50

Answer : 50/100 = 1/2

rahul16singh28 · Answer

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.

niks18 · Answer

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

rahul16singh28 · Answer

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.

Bunuel · Answer

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

niks18 · Answer

rahul16singh28 our problem is solved

EgmatQuantExpert · Answer

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

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