Let represent the greatest integer less than or equal to x. If n is a positive integer such that - = 2, what is the minimum possible value of n? A. 7 B. 8 C. 9 D. 12 E. 19

We can start with smallest numbers in the options: For n = 7: = 1, = 0, so – = 1 – 0 = 1 So 7 doesn’t work. For n = 8: = 2, = 0, so 2 – 0 = 2 8 satisfies the condition Answer: B

The best way would be to test the values, and just by looking at it, 8 seems to be a good contender. - = - = 2-0 = 2 Anything less than 8 will give an answer less than 2, and the other options will give something greater. Answer: n=8.

Checking the options: - 7 -> - = 1-0=1 - 8 -> - = 2-0=2 - 9 -> - = 2-1=1 - 12 -> - = 3-1=2 - 19 -> - = 4-2=2 8 is the minimum value that fits IMO B

The easiest way to do this question is to check each answer choice. We don't have to do full division to find the exact decimal representation. Fortunately, the numbers are small. Approximation is the way to go. (1). n=7 will be less than 2 and greater than 1, because 4*2=8 and 4*1 = 4, and 7 lies between 4 and 8. Decimal representation will be 1.xx Greatest integer less than or equal to 1.xx is 1. will be less than 1 and greater than 0 for the same reasons as . Decimal represesntation will be 0.xx Greatest integer less than or equal to 0.xx is 0 1-0 = 1. This is not the answer. (1). n=8 will be exactly 2, because 4*2=8. Greatest integer less than or equal to 2 is 2. will be less than 1 and greater than 0. Decimal represesntation will be 0.xx Greatest integer less than or equal to 0.xx is 0 2-0 = 2. This is correct Answer - B

Let’s try options A. n=7 = = 1, = =0==> 1-0=0 not matching Trying n=8, =2, =0,===> 2-0=2 n=8 is the minimum possible value satisfying the equation.

Go by the options in this question. A. 7 : - = 1- 0 = 1 B. 8 : - = 2- 0 = 2 C. 9 : - = 2- 1 = 1 D.12: - =3- 1 = 2 E.19: - = 4-2 = 2 Least number is 8, Option B.

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GMAT Club Olympics 2025 (Day 4): Let [x] represent the greatest intege

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We test the answer choices in order, examining how many whole times n fits into 4 and into 9 and then subtracting those two whole‐number results. When n is seven, it fits into four once and into nine not at all, giving a difference of one. When n is eight, it fits into four exactly twice but still fits into nine zero times, yielding a difference of two. All smaller values fail to produce a difference of two, so eight is the smallest integer that makes the greatest‐integer count for n divided by four exceed that for n divided by nine by exactly two. Thus the minimum possible value of n is eight.

Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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We can start with smallest numbers in the options:

For n = 7:
[7/4] = 1, [7/9] = 0, so [7/4] – [7/9] = 1 – 0 = 1
So 7 doesn’t work.

For n = 8:
[8/4] = 2, [8/9] = 0, so 2 – 0 = 2

8 satisfies the condition

Answer: B

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The best way would be to test the values, and just by looking at it, 8 seems to be a good contender.
[8/4]-[8/9] = [2]-[0.89] = 2-0 = 2

Anything less than 8 will give an answer less than 2, and the other options will give something greater.

Answer: n=8.

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Checking the options:
- 7 -> [7/4] - [7/9] = 1-0=1
- 8 -> [8/4] - [8/9] = 2-0=2
- 9 -> [9/4] - [9/9] = 2-1=1
- 12 -> [12/4] - [12/9] = 3-1=2
- 19 -> [19/4] - [19/9] = 4-2=2

8 is the minimum value that fits

IMO B

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Since n/9 = 0, for all n < 9, the difference simplifies to n/4.
For n less than 8 or n greater than or equal to 4, n/4=1,n/9=0 with a difference of 1.

To get difference = 2,
n/4 = 2, which first happens at n = 8.
Therefore, the minimum n satisfying the equation is 8.

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The easiest way to do this question is to check each answer choice. We don't have to do full division to find the exact decimal representation. Fortunately, the numbers are small. Approximation is the way to go.

(1). n=7
[7/4] will be less than 2 and greater than 1, because 4*2=8 and 4*1 = 4, and 7 lies between 4 and 8. Decimal representation will be 1.xx
Greatest integer less than or equal to 1.xx is 1.
[7/9] will be less than 1 and greater than 0 for the same reasons as [n/4]. Decimal represesntation will be 0.xx
Greatest integer less than or equal to 0.xx is 0

1-0 = 1. This is not the answer.

(1). n=8
[8/4] will be exactly 2, because 4*2=8.
Greatest integer less than or equal to 2 is 2.
[8/9] will be less than 1 and greater than 0. Decimal represesntation will be 0.xx
Greatest integer less than or equal to 0.xx is 0

2-0 = 2. This is correct

Answer - B

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We can start solving from answer choices. If n =7 then difference is 1 but if n=8 we get difference as 2
Hence Ans B

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On checking each of the options from A to E,

If n=7, n/4=7/4=1.75. The greatest integer value of 1.75 will be equal to 1 & n/9 = 7/9=0.777. The greatest integer value of 0.7777 is 0.

Therefore, the difference between the two is 1, which does not satisfy the condition. Hence (A) is incorrect

If n=8, n/4=8/4=2. The greatest integer value of 2 will be equal to 2 & 8/9 = 8/9=0.888. The greatest integer value of 0.8888 is 0.

Therefore, the difference between the two is 2, which satisfies the condition of the least value possible as all the other options subsequently are greater than 8. Hence (B) is correct

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Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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Let’s try options A. n=7
[n/4]=[7/4]= 1,
[n/9]=[7/9]=0==> 1-0=0 not matching

Trying n=8,
[8/4]=2,
[8/9]=0,===> 2-0=2 n=8 is the minimum possible value satisfying the equation.

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Go by the options in this question.
A. 7 : [7/4] - [7/9] = 1- 0 = 1
B. 8 : [8/4] - [8/9] = 2- 0 = 2
C. 9 : [9/4] - [9/9] = 2- 1 = 1
D.12: [12/4]-[12/9]=3- 1 = 2
E.19: [19/4]-[19/9]= 4-2 = 2

Least number is 8, Option B.

Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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Let me just test some numbers:
n = 7:

[7/4] = 1, [7/9] = 0
1 - 0 = 1 ❌

n = 8:

[8/4] = 2, [8/9] = 0
2 - 0 = 2 ✓

The minimum is 8.

Answer: B. 8

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Given,
[n/4] - [n/9] = 2

Let,
a = [n/4]
b = [n/9]
=> a - b = 2
=> a = b +2

Lets try to satisfy this condition 1 by 1 from the lowest given options,

A. n = 7

a = [7/4] = 1
b + 2 = [7/9] = 0 + 2 = 2
Doesent satisfy

B. n = 8

a = [8/4] = 2
b + 2 = [8/9] = 0 + 2 = 2
Satisfies and is the minimum option available for n

B. 8

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Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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With the [x] brackets meaning greatest integer less than or equal to x. I imagine that [n/9] is equal to 0. so that n/4 needs to equal 2 which would be 8.

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Best way is to put values for n, using the ans choice one by one, whichever smallest value gives us a 2 , thats the answer, using 8, [8/4]=2 , [8/9]=[0.smth]=0, 2-0=2 , Hence, B.

Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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For the given values,

7 ([7/4]-[7/9]=1-0=1) doesn't fit
and 8 ([8/4]-[8/9]=2-0=2) fits

so 8 must be the answer.

The right answer is B

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Trying choices from least to greater:

Max integer =<(7/4) - Max integer =<(7/9) = 1.75 -0.7777 equals => 1-0 = 1 (NO)
Max integer =<(8/4) - Max integer =<(8/9) = 2-0.88888 equals => 2-0 = 1 (YES)

Answer B

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Option B is the correct answer.

Lest understand the question and how we reach to our answer.

Fir the question tells us the "[x] represent the greatest integer less than or equal to x" which means that [2] = 2 and [3.5] = 3. Then it further tells us that n is a positive integer and [n/4] - [n/9] = 2, and asks us the minimal possible value of n.

Now lets try putting in the values available in the option in the above mentioned equation and check which one is our answer.

Option A: n = 7, From here we get [7/4] = 1 and [7/9] = 0, and if we put these value in the condition ([n/4] - [n/9])=>(1-0 = 1) then we find that 1 is not equal to 2. Eliminated

Option B: n = 8, From here we get [8/4] = 2 and [8/9] = 0, and if we put these value in the condition ([n/4] - [n/9])=>(2-0 = 2) then we find that it matches the answer. Selected

Option C: n = 9, From here we get [9/4] = 2 and [9/9] = 1, and if we put these value in the condition ([n/4] - [n/9])=>(2-1 = 1) then we find that 1 is not equal to 2. Eliminated

Option D: n = 12, From here we get [12/4] = 3 and [12/9] = 1, and if we put these value in the condition ([n/4] - [n/9])=>(3-1 = 2). Eliminated as it is not the smallest value of 'n'.

Option E: n = 19, From here we get [19/4] = 4 and [19/9] = 2, and if we put these value in the condition then we find that ([n/4] - [n/9])=>(4-2 = 2). Eliminated as it is not the smallest value of 'n'.

Now after check all the option we can see that Option B, Option D & Option E all three of them are giving 2 as the answer so which one to choose as the answer. Here if you check the question is asking for the minimum value of 'n' which satisfy the calculation which '8' i.e. Option B so on this basis can can also eliminate Option D & E as well.

Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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The Most important thing to understand is the nature of the function.

[x] represent the greatest integer less than or equal to x

if x= 4.9 or 4.5 or 4.1 then [x]= 4

Then it's just a matter of plugging in the numbers. Start with the smaller ones because we need the minimum possible value of n.
Start with 7.
Does not work.

Try 8
[8/4]-[8/9]=2?
[2]-[.89]=2?
2-0=2? YES

Answer is B)8

Bunuel

Let [x] represent the greatest integer less than or equal to x. If n is a positive integer such that [n/4] - [n/9] = 2, what is the minimum possible value of n?

A. 7
B. 8
C. 9
D. 12
E. 19

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It is given that $[x] <= x$ & $ [n/4] - [n/9] = 2$
We have to find the minimum possible value of $n$.

Let $[n/4] = a$; $[n/9] = b$,
we have $a - b = 2 $ ~ (1)
Now we know for the greatest integer function, $[x]<= x < [x] + 1$;
Substituting we get,
$a <= n/4 < a + 1$; this converts to,
$4a <= n < 4a + 4 $ ~ (2)

Also,
$b <= n/9 < b + 1$
$9b <= n < 9b + 9$ ~ (4)
From (1) we also have $a = b + 2$,

Substituting this in (3) we get,
$4b + 8 <= n < 4b + 8 + 4$
$4b + 8 <= n < 4b + 12 $~ (5)

Now we have two distinct equations for n in terms b; (4) & (5),
$9b <= n < 9b + 4$
$4b + 8 <= n < 4b + 12$

Now it is mentioned that $n$ is a positive integer so $b$ must be a non-negative integer.
To find the minimum possible value, we have to substitute the smallest possible value for $b$ that is non-negative, which is 0.

We have,
$9b <= n < 9b + 9$
$4b + 8 <= n < 4b + 12$, substituting $b = 0 $ we get,
$0 <= n < 9$
$8 <= n < 12$

The only values of n that satisfy both the equations should lie within the intersection of the intervals $[0, 9)$ & $[8, 12)$; That is the interval $8 <= n < 9$

Since n is an integer value, the only possible value that n can take is $8$.

This is the minimum possible value for n as substituting $b = 1$ would provide a significantly larger range with bigger values for $n$.

Therefore, the correct option is Option B

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I just back-solving for each choice for POE:
[7/4] - [7/9] = [1.75] - [0.abcd] = 1 - 0 = 1 - NOT the answer
[8/4] - [8/9] = [2] - [0.abcd] = 2 = winner winner chicken dinner!