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

asked: minimum value of n so that - = 2 Start by substitution from option A A. = 1 and = 0 then 1-0 is not equals to 2. Next option B. = 2 and = 0 then 2-0 is equal to 2 Hence Our answer since other options are greater than this value and we had to find the minimum.

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...... - = - = 1-0=1........No B. 8........ - = - = 1-0=1 or 2-0=2 ......Yes (as can be less than or equal to n) C. 9........ - = - =2-0=2 D. 12...... - = - =2-1=1 or 3-1=2 E. 19...... - = - =4-2=2 B (8) is minimum possible value of n

since, - =2 and all the options are positive integers. & both are positive. so the only way given equation will hold true for the least value of n, if =2 and =0 for =2 , 2<=n/4<3, n=(8,9,10,11) for =0, 0<=n/9<1, n=(0,1,2,3,4,5,6,7,8) the common value for n from both the sets is 8 So the least value n can take is 8.

Start plug-in the least value 7 and 8, Both of the options are eliminated because it doesn't satisfy the conditions given in the q-stem. Assuming the value 9, - = 2 - 1 = 1; Eliminated Assuming the value 12, - = 3 - 1 = 2; Satisfies Assuming the value 19, - = 4 - 1 = 1; Eliminated Option D Note: As given in the q-stem, it has to be the difference between the quotient value. A good tricky question

I have applied hit and trial method using the options given. Since question is asking the minimum possible value, I am starting with the lowest option which is 7. - = 1-0=1, Not correct Let's apply the second lowest option, which is 8. - = 2-0=2, Correct. Please note that the other options could also give 2 but since the question is asking the lowest value, I am going to stop here.

- = 2 used options to get to the answer. start with least : - 7 = 1 = 0 Therefore answer is 1 not correct - 8 = 2 = 0 There answer is 2 which is correct No need to go further.

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

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I think the fastest way is to solve this is to plug in numbers. We can think of [x] as a function removing all the decimal places of x.

For answer choice A:
[7/4] - [7/9] = 1-0 = 1

Thus, A is incorrect

For answer choice B:
[8/4] - [8/9] = 2 - 0 = 2

B is correct.

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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1. if n=7, result becomes 1-0=1
2. if n=8, result becomes 2-0=2..YES
3. n=9, 2-1=1
4.n=12, 3-1=2...YES

n=8 lowest

Ans B

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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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asked: minimum value of n so that [n/4] - [n/9] = 2

Start by substitution from option A
A. [7/4] = 1 and [7/9] = 0 then 1-0 is not equals to 2. Next option

B. [8/4] = 2 and [8/9] = 0 then 2-0 is equal to 2 Hence Our answer since other options are greater than this value and we had to find the minimum.

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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...... [7/4]-[7/9]=[1.75]-[0.78]= 1-0=1........No
B. 8........[8/4]-[8/9]=[2]-[0.89]= 1-0=1 or 2-0=2 ......Yes (as [n]can be less than or equal to n)
C. 9........[9/4]-[9/9]=[2.25]-[1]=2-0=2
D. 12......[12/4]-[12/9]=[3]-[1.3]=2-1=1 or 3-1=2
E. 19......[19/4]-[19/9]=[4.75]-[2.12]=4-2=2

B (8) is minimum possible value of n

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Test:
n = 7 => [n/4] - [n/9] = [7/4] - [7/9] = 1-0=1#2

n = 8 => [n/4] - [n/9] = [8/4] - [8/9] = 2-0 =2 => minimum possible value of n is 8.

We don’t need to test the rest of the options.

Answer: B. 8

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since, [n/4]-[n/9]=2 and all the options are positive integers.
[n/4] & [n/9] both are positive.
so the only way given equation will hold true for the least value of n,
if [n/4]=2 and [n/9]=0
for [n/4]=2 , 2<=n/4<3, n=(8,9,10,11)
for [n/9]=0, 0<=n/9<1, n=(0,1,2,3,4,5,6,7,8)
the common value for n from both the sets is 8
So the least value n can take is 8.

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Start plug-in the least value 7 and 8,

Both of the options are eliminated because it doesn't satisfy the conditions given in the q-stem.

Assuming the value 9, [9/4] - [9/9] = 2 - 1 = 1; Eliminated
Assuming the value 12, [12/4] - [12/9] = 3 - 1 = 2; Satisfies
Assuming the value 19, [19/4] - [19/9] = 4 - 1 = 1; Eliminated
Option D

Note: As given in the q-stem, it has to be the difference between the quotient value. A good tricky question

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starting with lowest options : 7/4 -7/9 = 1-0 = 1 not 2. then, 8/4-8/9 = 2-0 =2 hence B .

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I have applied hit and trial method using the options given.
Since question is asking the minimum possible value, I am starting with the lowest option which is 7.
[7/4] - [7/9] = 1-0=1, Not correct
Let's apply the second lowest option, which is 8.
[8/4] - [8/9] = 2-0=2, Correct.
Please note that the other options could also give 2 but since the question is asking the lowest value, I am going to stop here.

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

used options to get to the answer.

start with least :

- 7
[7/4] = 1
[7/9] = 0

Therefore answer is 1 not correct

- 8
[8/4] = 2
[8/9] = 0

There answer is 2 which is correct

No need to go further.

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[x] <= x

This means [n/4] <= n/4 and [n/9]<=n/9
This can be solved quickly by directly substituting values from choices

If n = 7
[n/4] = 1
[n/9] = 0
So difference is 1 and not equal to 2

If n=8
[n/4] = 2
[n/9] = 0
[n/4] - [n/9] = 2 as given in question
Hence n=8 is minimum number satisfying the equation

Option B is correct

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 question is best solved by substituting the values,

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

n=8,12,19 satisfies the condition, and minimum value of n is thus 8

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n> 0,
So for 0<n<9, [n/9] =0.
=> [n/4]= 2 which means n>=8

Since we need to find min. n. n=8

B is the answer.

Alternatively, in question types of min or max possible values, start by putting values from B and D. you can easily get to answer in 3 trials.

putting x=8, [8/4] -[ 8/9] = 2-0 = 2 Ok,
try x=7 now [7/4] - [7/9] = 1-0 =1 so B is the answer

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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Lets back solve this using options:

Option A:
[7/4] - [7/9] = 1 - 0 = 1

Option B:
[8/4] - [8/9] = 2 - 0 = 2 [Correct]

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 easier to solve this problem by using the options and substituting them in place of n.

Option A: if n = 7, the expression would be = 1. Not the correct answer.
Option B: if n = 8, the expression would be = 2 which is what we want.
Since all other options are greater than option B, option B is the correct answer.

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 quickest way to solve this kind of problem is to put options in the solution

n | n/4 | n/9. | Difference

7 | 1 | 0 | 1
8 | 2 | 0 | 2 -> SATISFIES

Question ask the minimum possible values of n

8 is the answer. | 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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two considerations
- [X] must be an interger
- [X] is rounded down and not up

with that in mind, [n/4] - [n/9] can result to two if [n/4] = 2 and [n/9] = 0 or if [n/4] = 3 and [n/9] = 1

we see n needs a larger value for [n/4] to result to three than to two ( take 12/4 gives 3 and 8/4 gives 2 )

so we want to minimize the value of n such that n/4 should be greater than or equal to 2 but less than 3.

the solution is 8 ( B)

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?

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 function is simple. [x] is the greatest integer <= x. Means remove the decimals.
The trap is many numbers can fulfill this equation. and the hint is minimum possible value is n, which means answers may have more than one satisfying solutions. So, we have choose the minimum among them.
Lets check from low values,

A. if n= 7, [7/4] = 1, [7/9] = 0. 1-0=2 is wrong. so not the answer.
B. if n= 8, [8/4] = 2, [8/9] = 0. 2-0=2 is correct and this is the lowest value among options.

Hence answer is (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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Pretty simple, should be B

[8/4]=2
[8/9]=0

Thus, 2-0=2, which satsifies the equation. Hence, 8 is the smallest

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given,

[n/4] - [n/9] = 2
minimum possible value of n = ?

lets consider a = n/4 , b = n/9

a-b = 2
a = 2 + b ----> equation 1
we know floor function |a| <= a <|a|+1 ----> condition
(<= - less than or equal to)

substitute a and b in condition

we obtain following :

1) 4a <= n < a+1
2) 9b <= n < 9b + 9

based on equation 1, simplify above statements for n

we obtain,

1) 4b + 8 <= n < 4b+ 12
2) 9b <= n < 9b + 9

now, lets find the minimum positive integer n
lets substitute b = 0 in above equations
1) 8 <= n < 12
2) 0 <= n < 9

we find the intersection at 8 from above statements , Therefore, the minimum value = 8

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