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

The number n (the minimum possible one) must be a number such as =2 and it must be less than 9 so =0. The number 8 meets the requirements. Correct answer is B

If n = 7, then we have - = 1. if n=9, - = again, 1. thus, the minimum possible value of n will be D. 12 , because - = 2.

Utilised answer choices here; A: for n=7; will give 1 and will give 0 so 1-0=1 => not valid as we wanted the difference to be 2 B: for n=8; will give 2 (as the function can be equal to the number inside) and will give 0 so 2-0=2 => correct ans as other numbers are higher in magnitude and que asked about the minimum

We can start working with the options, (a) 7/4 = 1.75 so \frac{ } = 1 7/9 = 0.77 so \frac{ } = 0 1-0\neq{2} (b) 8/4 = 2 so \frac{ } = 2 8/9 = 0.88 so \frac{ } = 0 2-0 =2. Hence (b)

is function such that it will output greatest integer less than or equal to x, For example which 2.5 but function will give greatest integer less than or equal to 2.5 which is 2. Now for - = 2, its best to try options. Option A: which 1 point something will be 1 and is 0 point something which will be 0 As, 1 - 0 is not equal to 2. Hence INCORRECT. Option B: which exactly 2 and is 0 point something which will be 0 As, 2 - 0 is equal to 2. Hence CORRECT.

Let the given equation be − =2. We are looking for the minimum positive integer value of n that satisfies this equation. Let A= and B= . So, A−B=2, which means A=B+2. By the definition of the greatest integer function:A≤n/4<A+1⟹4A≤n<4A+4B≤n/9<B+1⟹9B≤n<9B+9 Substitute A=B+2 into the first inequality:4(B+2)≤n<4(B+2)+44B+8≤n<4B+12 Now we have two inequalities for n: 9B≤n<9B+9 4B+8≤n<4B+12 For a common range for n to exist, there must be an overlap between these two intervals. The lower bound for n is max(9B,4B+8). The upper bound for n is min(9B+9,4B+12). We need to find the smallest positive integer n. Let's test values for B (which must be a non-negative integer since n is positive). Case 1: B = 0A=0+2=2. From A= , we have 2≤n/4<3⟹8≤n<12. From B= , we have 0≤n/9<1⟹0≤n<9. For n to satisfy both, we need 8≤n<9. The only integer in this range is n=8. Let's check n=8: − =2−0=2. This satisfies the equation. So n=8 is a possible value. Since we started with the smallest possible B and found an integer n, this n is the minimum possible value. Let's quickly check other values of B to ensure n=8 is indeed the minimum.Case 2: B = 1A=1+2=3. From A= , we have 3≤n/4<4⟹12≤n<16. From B= , we have 1≤n/9<2⟹9≤n<18. For n to satisfy both, we need max(12,9)≤n<min(16,18), so 12≤n<16. Possible integer values for n are 12,13,14,15. All of these are greater than 8. Case 3: B = 2A=2+2=4. From A= , we have 4≤n/4<5⟹16≤n<20. From B= , we have 2≤n/9<3⟹18≤n<27. For n to satisfy both, we need max(16,18)≤n<min(20,27), so 18≤n<20. Possible integer values for n are 18,19. All of these are greater than 8. Since we found n=8 for the smallest possible value of B=0, and subsequent values of B lead to larger n, the minimum possible value of n is 8. The final answer is 8

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

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The number n (the minimum possible one) must be a number such as [n/4]=2 and it must be less than 9 so [n/9]=0.

The number 8 meets the requirements.

Correct answer is B

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If n = 7, then we have [1] - [0] = 1.

if n=9, [2] - [1] = again, 1.

thus, the minimum possible value of n will be D. 12, because [3] - [1] = 2.

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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Because of the condition less than or equal to x we can straight away test the answers from the smallest which is 7
So 7/4 = 1.75 which equals 1 while 7/9= 0.77 which equals 0 so 1-0= 1 which is incorrect
For 8 8/4 = 2 and 8/9 =0.88 which equals 0 so 2-0= 2 so min value of n is 8
ANS 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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Calculate the value of the expression for all the values in the options in ascending order. The first one that is valid is the answer.
A. [7/4]-[7/9]=1-0=1 not equal to 2
B. [8/4]-[8/9]=2-0=2

Answer B

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Utilised answer choices here;
A: for n=7; [7/4] will give 1 and [7/9] will give 0 so 1-0=1 => not valid as we wanted the difference to be 2
B: for n=8; [8/4] will give 2 (as the function can be equal to the number inside) and [8/9] will give 0 so 2-0=2 => correct ans as other numbers are higher in magnitude and que asked about the minimum

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We can start working with the options,

(a) 7/4 = 1.75 so $\frac{[7}{4]}$ = 1
7/9 = 0.77 so $\frac{[7}{9]}$ = 0
$1-0\neq{2}$

(b) 8/4 = 2 so $\frac{[7}{4]}$ = 2
8/9 = 0.88 so $\frac{[7}{9]}$ = 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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[x] is function such that it will output greatest integer less than or equal to x,
For example [5/2] which 2.5 but function will give greatest integer less than or equal to 2.5 which is 2.

Now for [n/4] - [n/9] = 2, its best to try options.

Option A: [7/4] which 1 point something will be 1 and [7/9] is 0 point something which will be 0
As, 1 - 0 is not equal to 2. Hence INCORRECT.

Option B: [8/4] which exactly 2 and [8/9] is 0 point something which will be 0
As, 2 - 0 is equal to 2. Hence CORRECT.

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Let the given equation be [n/4]−[n/9]=2.
We are looking for the minimum positive integer value of n that satisfies this equation.

Let A=[n/4] and B=[n/9].
So, A−B=2, which means A=B+2.

By the definition of the greatest integer function:A≤n/4<A+1⟹4A≤n<4A+4B≤n/9<B+1⟹9B≤n<9B+9

Substitute A=B+2 into the first inequality:4(B+2)≤n<4(B+2)+44B+8≤n<4B+12

Now we have two inequalities for n:

9B≤n<9B+9

4B+8≤n<4B+12

For a common range for n to exist, there must be an overlap between these two intervals.
The lower bound for n is max(9B,4B+8).
The upper bound for n is min(9B+9,4B+12).

We need to find the smallest positive integer n. Let's test values for B (which must be a non-negative integer since n is positive).

Case 1: B = 0A=0+2=2.
From A=[n/4], we have 2≤n/4<3⟹8≤n<12.
From B=[n/9], we have 0≤n/9<1⟹0≤n<9.

For n to satisfy both, we need 8≤n<9. The only integer in this range is n=8.
Let's check n=8:[8/4]−[8/9]=2−0=2.
This satisfies the equation. So n=8 is a possible value. Since we started with the smallest possible B and found an integer n, this n is the minimum possible value.

Let's quickly check other values of B to ensure n=8 is indeed the minimum.Case 2: B = 1A=1+2=3.
From A=[n/4], we have 3≤n/4<4⟹12≤n<16.
From B=[n/9], we have 1≤n/9<2⟹9≤n<18.
For n to satisfy both, we need max(12,9)≤n<min(16,18), so 12≤n<16.
Possible integer values for n are 12,13,14,15. All of these are greater than 8.

Case 3: B = 2A=2+2=4.
From A=[n/4], we have 4≤n/4<5⟹16≤n<20.
From B=[n/9], we have 2≤n/9<3⟹18≤n<27.
For n to satisfy both, we need max(16,18)≤n<min(20,27), so 18≤n<20.
Possible integer values for n are 18,19. All of these are greater than 8.

Since we found n=8 for the smallest possible value of B=0, and subsequent values of B lead to larger n, the minimum possible value of n is 8.

The final answer is 8

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