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

Correct Answer= Option B, 8 Check by option approach, A. 7 If we keep n=7 - = 2 - = 2 This case have difference somewhere around 1. Eliminate this option. B. 8 If we keep n=8 - = 2 - = 2 This case have difference somewhere around 2. As we have asked minimum value of n so this option is the closest option.

Given : = 1; = 1; = 1 Using POE(Process of Elimination) opt A: - = 1 - 0 = 1 opt B: - = 2 - 0 = 2 Hence opt B

Substitute each value starting from the smallest. - = 1-0 = 1 Incorrect - = 2-0 = 2, correct Thus B.

Given, (i) represent the greatest integer less than or equal to x (ii) N > 0 (iii) - = 2 Minimum possible value of n =? Checking the option, a. - = 2 – 1 = 1 b. - = 2 – 1 = 1 c. - = 3 – 1 = 2 d. - = 3 – 2 = 1 e. - = 5 – 3 = 2 C & E satisfy the given equation in info (ii), Possible values among options are 9 & 19, Minimum value is 9. Ans : C

we are given greatest integer function of n/4 minus greatest integer of n/9 equals 2 so \ − \ = 2 we want to find the smallest positive integer n such that this holds try small values of n: n = 7 \ = 1.75 → 1 \ = 0.77 → 0 1 − 0 = 1 → not valid n = 8 \ = 2 \ = 0 2 − 0 = 2 → valid check if this is the smallest n = 1 to 7 don’t satisfy, already checked 7 gave 1 so minimum value is 8 final answer is b

<= x (It showcases that the value of x should not be greater than x) such as, = 2 and = 2 indicates the Greatest Integer Function - = 2 A) 7 = - < 2 (Not possible) B) 8 = - = 1 < 2 (not possible) C) 9 = - = 2.25 - 1 = = 1 < 2 (not possible) D) 12 = - = 3 - 1.33 = = 2 = 2 (possible) E) 19 = - = - = 5 - 2 = 3 (not possible)

We need to find the minimum value of 'n' - = 2 Let's go by the answer choices on this one A. 7 - = 2 => 1.x - 0.y = 2 => 1 - 0 = 2 Does not satisfy B. 8 - = 2 => 2 - 0.y = 2 => 2 - 0 = 2 Satisfies C. 9 - = 2 => 1.x - 1 = 2 => 1 - 1 = 2 Does not satisfy D. 12 - = 2 => 3 - 1.y = 2 => 3 - 1 = 2 Satisfies E. 19 - = 2 => 4.x - 1.y = 2 => 4 - 1 = 2 Does not satisfy So, 8 and 12 satisfy the condition but we need the minimum possible value of n hence we go with 8 Option B

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

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Correct Answer= Option B, 8
Check by option approach,
A. 7
If we keep n=7
[n/4]-[n/9]= 2
[7/4]-[7/9]= 2
This case have difference somewhere around 1. Eliminate this option.
B. 8
If we keep n=8
[n/4]-[n/9]= 2
[8/4]-[8/9]= 2
This case have difference somewhere around 2. As we have asked minimum value of n so this option is the closest option.

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This one i just trial-and-errored my way.

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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Given : [1.999] = 1; [1.1] = 1; [1.0] = 1
Using POE(Process of Elimination)
opt A: [7/4] - [7/9] = 1 - 0 = 1
opt B: [8/4] - [8/9] = 2 - 0 = 2
Hence opt 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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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

Substitute each value starting from the smallest.

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

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

Thus B.

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Given,
(i) [x] represent the greatest integer less than or equal to x
(ii) N > 0
(iii) [n/4] - [n/9] = 2
Minimum possible value of n =?

Checking the option,
a. [7/4] - [7/9] = 2 – 1 = 1
b. [8/4] - [8/9] = 2 – 1 = 1
c. [9/4] - [9/9] = 3 – 1 = 2
d. [12/4] - [12/9] = 3 – 2 = 1
e. [19/4] - [19/9] = 5 – 3 = 2
C & E satisfy the given equation in info (ii),
Possible values among options are 9 & 19,
Minimum value is 9.
Ans : C

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 are given
greatest integer function of n/4 minus greatest integer of n/9 equals 2
so \[n/4] − \[n/9] = 2

we want to find the smallest positive integer n such that this holds

try small values of n:

n = 7
\[7/4] = 1.75 → 1
\[7/9] = 0.77 → 0
1 − 0 = 1 → not valid

n = 8
\[8/4] = 2
\[8/9] = 0
2 − 0 = 2 → valid
check if this is the smallest

n = 1 to 7 don’t satisfy, already checked 7 gave 1
so minimum value is 8

final answer is b

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[x] <= x (It showcases that the value of x should not be greater than x)
such as,
[2.1] = 2 and [1.7] = 2
[ ] indicates the Greatest Integer Function

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

A) 7 = [7/4] - [7/9 ] < 2 (Not possible)

B) 8 = [8/4]- [8/9] = 1 < 2 (not possible)

C) 9 = [9/4] - [9/9] = 2.25 - 1 = [1.25] = 1 < 2 (not possible)

D) 12 = [12/4] - [12/9] = 3 - 1.33 = [1.7] = 2 = 2 (possible)

E) 19 = [19/4] - [19/9] = [4.75] - [2.11] = 5 - 2 = 3 (not possible)

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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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We need to find the minimum value of 'n'

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

Let's go by the answer choices on this one

A. 7
[7/4] - [7/9] = 2
=> 1.x - 0.y = 2
=> 1 - 0 = 2
Does not satisfy

B. 8
[8/4] - [8/9] = 2
=> 2 - 0.y = 2
=> 2 - 0 = 2
Satisfies

C. 9
[9/4] - [9/9] = 2
=> 1.x - 1 = 2
=> 1 - 1 = 2
Does not satisfy

D. 12
[12/4] - [12/9] = 2
=> 3 - 1.y = 2
=> 3 - 1 = 2
Satisfies

E. 19
[19/4] - [16/9] = 2
=> 4.x - 1.y = 2
=> 4 - 1 = 2
Does not satisfy

So, 8 and 12 satisfy the condition but we need the minimum possible value of n hence we go with 8

Option B

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We can do this by option substitution. Taking median value 9, so n/4 will come as 2 because less or equal to x. And n/9 would come 1. Difference would be 1. Going to higher value 12 we get n/4 as 3 and n/9 as 1 so this satisfies the condition. We cant take 7 because n/4 is 1 then. Option D-12

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

A. n = 7
=> 7/4 = 1. ...
=> [7/4] = 1
=> 7/9 = 0. ...
=> [7/9] = 0
=> 1 - 0 = 1 =/= 2 Hence, Eliminated

B. n = 8
=> 8/4 = 2
=> [8/4] = 2
=> 8/9 = 0. ...
=> [8/9] = 0
=> 2 - 0 = 2 Hence, Correct

Since we found minimum value of n, we do not need to check the other values.

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One can substitute the value for n in the equation- [n/4]-[n/9]=2
For 7: [n/4]= 1 and [n/9]=0, not possible.
For 8: [n/4]=2 and [n/9]=0, possible.

Therefore the minimum value of n is 8.

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greedy approach draw table for each options against the values of equation

(option) .... [n/4] .... [n/9] ... eq value

7 ... 1... 0 ...1
8 ... 2 ... 0 ... 2 < contender
(other options are larger, and we are asked to find minimum)

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We will test the smallest value of n that satisfies the equation –
n = 7 ;
[7/4] – [7/9] = 1 – 0 = 1 – not satisfied
N = 8 ;
[8/4]-[8/9] = 2- 0 = 2 –satisfied
Ans – 8 [B]

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Greatest integer less than or equal to x.

Let's understand using two examples [4] = 4 & [3.5] = 3.

Now we understand, let's apply in the question [n/4] - [n/9] = 2. Let's put 8 -> [8/4] - [8/9] = [2] - [0.1..] = 2 - 0 = 2

8 is the minimum possible. (B)

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

Start from top
A 7 - 1-0 =1 - Incorrect
B 8 - 2 - 0 =2 - Correct.

Answer B .

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for minimum possible value of integer [n/9] =0 ; [n/4] = 2

option b satisfy the condition

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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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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According to the greatest integer function: [x]= greatest integer value <=x
Eg: x=2.8
[2.8]= 2

Now, let's use the trial and error method to find the minimum value of n for the expression: [n/4] - [n/9] =2

Plug in values of n from the options:
n=7
[7/4] = 1
[7/9]= 0
=> 1-0 != 2
Hence eliminate

n=8
[8/4] = 2
[8/9]= 0
=> 2-0 =2
Keep this option

Rest all options have value of n > 8 and we need to find the minimum value of n.

Hence,
ANSWER: B. 8

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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?

What is the minimum possible positive value that can be substituted into this expression [n/4] - [n/9] = 2 that would make the value 2.

Using 8

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

We get 2-0.888

Which is approximately 2.

Hence 8 satisfies this expression

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B. 8

Solve this by substituting options
for n=7
[7/4] - [7/9] = 1 - 0 = 1 --> incorrect
for n=8
[8/4] - [8/9] = 2 - 0 = 1 --> correct

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