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

GMAT Club Official Explanation: Notice that as n increases, both and increase, but increases faster. Every time n reaches a multiple of 4, increases by 1. Similarly, increases by 1 at multiples of 9. For example, for values n = 1 to 3, both n/4 and n/9 are less than 1, so and are both 0. When n reaches 4, becomes 1 while remains 0, giving a difference of 1. Now, for a difference of 2, we need a value of n for which is 2 while is still 0. Checking the next multiple of 4, which is 8, we get = 2 and = 0, giving the required difference of 2. So the minimum value of n is 8. Answer: B.

represent the greatest integer less than or equal to x Given, - = 2 has to be at least 2, i.e n has to be at least 8 Starting from there - - 2- 2-0 2 Answer 8

The best way would be to try the answer options starting from the minimum value of n from the answer choices. If n = 7 - = 1 - 0 = 1. Not the answer If n = 8 - = 2 - 0 = 2. This is our answer. Since we got the least value of n, other option choices need not be considered. Therefore, Option B

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; Incorrect B. 8 : - = 2 - 0 = 2; Correct C. 9 D. 12 E. 19 There is no need to evaluate other answer choice since we had to find minimum possible value of n. IMO B

Here I tried to fit the options that are given to find the solution Since is the greater integer less than equal to x that me if it comes in decimal like then it will come out as 1 By consider the above understanding, I see 8 can fit the best in the equation so - = - = 2-0=2. You can try others but this 8 is closest so my bet is solution B

Information given: - is greatest integer less than or equal to x (floor function) - We want a positive integer n, such that: - = 2 Question: - What is the minimum positive integer n that satisfies: - = 2 Solution: - = + 2 - A: 7, gives = 1.75 = 1, = = 0, difference = 1 - 0 = 0, too small - B: 8, gives = 2, = 0, difference = 2 - 0 = 0, valid - Since we want the minimum value, 8 is right Answer: B, 8

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? - = 2; = + 2; The Minimum Value of can be 2, since n is a positive integer. Then the Minimum value of n would be 8; Put 8 into the equation: - = 2 - 0 = 2;

We need to find the smallest value of n where - = 2. Remember, means "the greatest integer less than or equal to x." For example, = 3 and = 2 8/9 - 0.889.., so - = 2, and we need the minimum possible value, the answer is 8. The answer is B.

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

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Bunuel

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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

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Bunuel

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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

GMAT Club Official Explanation:

Notice that as n increases, both [n/4] and [n/9] increase, but [n/4] increases faster. Every time n reaches a multiple of 4, [n/4] increases by 1. Similarly, [n/9] increases by 1 at multiples of 9. For example, for values n = 1 to 3, both n/4 and n/9 are less than 1, so [n/4] and [n/9] are both 0. When n reaches 4, [n/4] becomes 1 while [n/9] remains 0, giving a difference of 1.

Now, for a difference of 2, we need a value of n for which [n/4] is 2 while [n/9] is still 0. Checking the next multiple of 4, which is 8, we get [8/4] = 2 and [8/9] = 0, giving the required difference of 2. So the minimum value of n is 8.

Answer: B.

General Discussion

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The key here is to remember that 0 is also an integer.

Thus, through trial and error:
A: 7/4 - 7/9 -> 1 - 0 = not 2, incorrect
B: 8/4 - 8/9 -> 2 - 0 = 2, correct

Answer: B) 8

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

Given, [n/4] - [n/9] = 2
[n/4] has to be at least 2, i.e n has to be at least 8 Starting from there

[n/4] - [n/9]
[8/4] - [8/9]
2-[0.x]
2-0
2

Answer 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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The best way would be to try the answer options starting from the minimum value of n from the answer choices.

If n = 7

[n/4] - [n/9] = 1 - 0 = 1. Not the answer

If n = 8

[8/4] - [8/9] = 2 - 0 = 2. This is our answer.

Since we got the least value of n, other option choices need not be considered.

Therefore, Option B

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B.

I thought of just putting options in question. Thus, option B looks great.

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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 - 0 = 1; Incorrect
B. 8 : [8/4] - [8/9] = 2 - 0 = 2; Correct
C. 9
D. 12
E. 19

There is no need to evaluate other answer choice since we had to find minimum possible value of n.

IMO B

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Here I tried to fit the options that are given to find the solution

Since [x] is the greater integer less than equal to x that me if it comes in decimal like [1.4] then it will come out as 1

By consider the above understanding, I see 8 can fit the best in the equation so [8/4]-[8/9] = [2]-[0.7]= 2-0=2.

You can try others but this 8 is closest so my bet is solution 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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Information given:
- [x] is greatest integer less than or equal to x (floor function)
- We want a positive integer n, such that: [n/4] - [n/9] = 2

Question:
- What is the minimum positive integer n that satisfies: [n/4] - [n/9] = 2

Solution:
- [n/4] = [n/9] + 2

- A: 7, gives [7/4] = 1.75 = 1, [7/9] = [0.777] = 0, difference = 1 - 0 = 0, too small
- B: 8, gives [8/4] = 2, [8/9] = 0, difference = 2 - 0 = 0, valid
- Since we want the minimum value, 8 is right

Answer: 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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Let [n/4] = x
[n/9] = y

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

Minimum value of x can be 2
Value of y when x=2 is 0 for equation to be valid

n= 8 satisfies this condition

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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?

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

[n/4] = [n/9] + 2;

The Minimum Value of [n/4] can be 2, since n is a positive integer.

Then the Minimum value of n would be 8;

Put 8 into the equation: [8/2] - [8/9] = 2 - 0 = 2;

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We need to find the smallest value of n where [n/4] - [n/9] = 2.
Remember, [x] means "the greatest integer less than or equal to x." For example, [37] = 3 and [5) = 5.
Let's try some values of n and see what happens:
When n = 8:
8/4 = 2, so [8/4] = 2
8/9 - 0.889.., so [8/91=0
2- 0=2/ When n = 9:
9/4 = 2.25, so [9/41 = 2
9/9 = 1, so [9/91=1
2-1=1 (not equal to 2)
Since n= 8 gives us [n/4] - [n/9] = 2, and we need the minimum possible value, the answer is 8.
The answer is B.

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In this lets check the options directly as it will be faster. lets start from the middle.

n = 9
n/4 = 2.25, so [n/4] =2
n/9 = 1, so [n/9] = 1
so 2-1 = 1 so not the ans lets move

n=8
n/4 = 2, so [n/4] = 2
n/9 = 0.88, so [n/9] = 0
so 2-0 = 2, so this can be the ans but lets check for 7 just to make sure

n=7
n/4=1.7 so [n/4] = 1
n/9 = 0.7 so [n/9] = 0
so 1-0 = 1 so not the ans.

Now even if we get 2 for other ans we are asked min value of. n so Ans = 8
Hence 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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Substitute the values in the choices in the equation we find that the minimum possible value of n is 12

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given that [x] represent the greatest integer less than or equal to x.
so [n/4] - [n/9] = 2, minimum value of n

when n is 7 ; [7/4]-[7/9] ; [1.75]-[.77] ; 2-1 =1 incorrect
when n is 8 ; [8/4]-[8/9] ; [2] - [ 0.88] ; 3-1 = 2 correct

OPTION B , 8 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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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

This question was provided by GMAT Club
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Did with hit and trial method with options

for 7 , [n/4] = 1 , [n/9] = 0 difference is 1 so not correct
for 8 , [n/4] =2 , [ n/9] = 0 difference is 2 and since least value of n is asked 8 is correct (B)

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I did it by putting the value of n from the ans options in the ques.

1) Putting 7 in [n/4] - [n/9] = 2

[7/4] - [7/9] = [1.75] - 0 = 1-0 = 1 Not correct

we have to round down, hence 1.75 rounded down is 1 and 7/9 would be 0.something, which will be rounded down to 0.

2) Putting 8, we have
[8/4] - [8/9] = 2-0 = 2 sufficient

Ans 8

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

Directly solving from options

let n=7

[7/4] - [7/9] = [1.75] - [0.77]= 1-0=1
Doesn't satisfy

n=8
[8/4] - [8/9] = [2] - [0.88]= 2-0=2

Rest values are greater than 8.

So the min value of n is 8. Correct answer is B.

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

Let's break it down with examples:
[2.2] = 2
[1.2] = 1
[2] = 2

Also, the question asks for the minimum possible possible number.
So, looking at it:
Even though 8, 12, and 19 all fit the final result,
only 8 is the minimum possible

n	n/4	【n/4】	n/9	【n/9】	【n/4】-【n/9】
7	1.75	1	0.77	0	1
8	2	2	0.88	0	2
9	2.25	2	1	1	1
12	3	3	1.33	1	2
19	4.75	4	2.11	2	2

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I just started plugging in the options.
Option A
[7/4]-[7/9]=2 ?
[1.something] -[0.something] which gives us 1-0 and that is not equal to 2.So this is wrong

Option B
[8/4]-[8/9]=2 ?
[2]-[0.something] which gives us 2-0 and that is equal to 2.This is right and this looks like the smallest integer in the options.

Check out the concept of greatest integer value for this and also note how negative values work in greatest integer value too.

So the 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

This question was provided by GMAT Club
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