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

Let's test from options ⌊n/4⌋−⌊n/9⌋=2 small positive integers for n: For n=7: ⌊7/4⌋=1, ⌊7/9⌋=0 1−0=1 ( small) For n=8: ⌊8/4⌋=2, ⌊8/9⌋=0 2−0=2 satisfies. Thus, the minimum positive integer n satisfying the equation is 8. B) 8

A = 7 = 1 = 0 1 - 0 = 1 Eliminate B = 8 = 2 = 0 2 - 0 = 2 Option B

The problem asks for the smallest positive integer n that satisfies the equation - = 2, where is the greatest integer less than or equal to x. The most direct way to solve this is to test the answer choices, starting with the smallest value. A. n = 7 - = - = 1 - 0 = 1. This is not equal to 2. B. n = 8 - = - = 2 - 0 = 2. This satisfies the equation. Since the options are listed in increasing order and n=8 is the first value to satisfy the condition, it is the minimum possible value.

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

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I have no idea here.....

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Best way is the option testing method.

Option A: 7
7 ÷ 4 = 1.75 → whole number part = 1
7 ÷ 9 = 0.77 → whole number part = 0
1 − 0 = 1 Not Correct

Option B: 8
8 ÷ 4 = 2 → whole number part = 2
8 ÷ 9 = 0.88 → whole number part = 0
2 − 0 = 2 Correct

Since we need the smallest possible number and the other options are greater than 8, it is not essential to check other options.

BunuelLet [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 test from options ⌊n/4⌋−⌊n/9⌋=2 small positive integers for n:

For n=7:
⌊7/4⌋=1, ⌊7/9⌋=0
1−0=1 ( small)
For n=8:
⌊8/4⌋=2, ⌊8/9⌋=0
2−0=2 satisfies.

Thus, the minimum positive integer n satisfying the equation is 8.

B) 8

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All the answers are positive. Since [n/4] > [n/9], and [n/4] - [n/9] = 2, we know that [n/4] must be >=2

The minimum number qualified for the above is 8. Testing with 8 as n, we see that 8 works.

Answer: 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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[n/4] and [n/9] are positive integers.

Hence, the minimum value of n/9 can be 0

Eliminate C, D, and E.

Try B

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

Hence, Option 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

I would like to solve this problem by substitution method

A. When n=7, [7/4]-[7/9]=1.7-[0.xx]=1.7-0=1.7
B When n=8 [8/4]-[8/9]=2-[0.xx]=2-0=2

All other values will give greater than 2 or equal to 2, since 8 is the minimum possible, answer is B

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We can substitute the values from the smallest in the given equation.

option 1- 7. If 7 is substituted the answer will come as 1. So it is incorrect

Option 2- 8. If a is substituted then equation will become[8/4]-[8/9]=2-0=2

Therefore 8 will be answer

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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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A = 7

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

1 - 0 = 1

Eliminate

B = 8

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

2 - 0 = 2

Option B

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N is a positive integer , and by solving the given equation, we get the minimum value of n is 8.

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The problem asks for the smallest positive integer n that satisfies the equation [n/4] - [n/9] = 2, where [x] is the greatest integer less than or equal to x.

The most direct way to solve this is to test the answer choices, starting with the smallest value.

A. n = 7

[7/4] - [7/9] = [1.75] - [0.77...] = 1 - 0 = 1. This is not equal to 2.

B. n = 8

[8/4] - [8/9] = [2] - [0.88...] = 2 - 0 = 2. This satisfies the equation.

Since the options are listed in increasing order and n=8 is the first value to satisfy the condition, it is the minimum possible value.

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Ans (B)

Plug in the values given in the answer

for 7
we get 1- 0 = 1 and not equal to 2

for 8
we get 2-0 = 2, satisfies

For 9
we get 2-1 not equal to 2

for 12
we get 3-1 = 2, satisfies

for 19
we get 4-2 = 2, satisfies

Howver, since the question stems asks us to find the minimum value we will choose B which is 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 notation [x] means to take the greatest integer that is less than or equal to x. For example, [4.9] is 4, and is 5. The problem asks for the smallest positive integer n that makes the equation [n/4] - [n/9] = 2 true.

The fastest way to solve this is to test the answer choices, starting with the smallest, since we are looking for the minimum value.

Try n = 7:
[7/4] - [7/9] = [1.75] - [0.77] = 1 - 0 = 1. This is not 2.

Try n = 8:
[8/4] - [8/9] = - [0.88] = 2 - 0 = 2. This matches the equation.

Since we started with the smallest choice and found the answer, 8 is the minimum possible value of n.

Thus, the correct 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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I would recommend using step by step substitution to get the correct answer

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

Option 2 : 8
Putting values : [8/4] - [8/9] = 2 - 0 = 2

Hence, answer 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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Greatest Integer function as per the give logic means the output would be integer less than or equal to x.

Example [0.9]=0 , [1.8]=1

Hence Now the question must be approached via the options.

A. 7 : [7/4] - [ 7/9] = 0 - 0 = 0 False not equal to RHS

B 8 : [8/4] - [8/9] = 2- 0 = 0 - True equal to RHS.

Similarly C and D is not sufficient.

Hence the answer is B

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$[\frac{n}{4}]- [\frac{n}{9}]=2$

Since the result is a positive integer, $[\frac{n}{4}]\geq{2}$

$n\geq{8}$

Answer: 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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Let [x] denote the Greatest Integer Function less than or equal to x

If n is a positive integer >0 , such that

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

To find: Minimum possible value of n ?

let’s substitute n = 7, [7/4] - [7/9] = 1.xyz - 0.abc = 1-0 = 1 . Eliminated.

when n = 8, [8/4] - [8/9] = 2 - 0.abc = 2-0 = 2 . Correct

Rest other values of n are greater than 8, so the minimum possible value of n = 8

Option B

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Find the minimum value of n, so we can try all value from smallest:
A. 7 => [n/4] - [n/9] = 1-0 = 1 >< 2 => Not correct
B. 8 => [n/4] - [n/9] = 2-1 = 2 => Correct. We don't need to try other bigger values because even that integer lead to [n/4] - [n/9] = 2, it still bigger than 8
=> 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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Ans: 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?

lets start with the options by putting them into the given equation.
A. 7 = [7/4] - [7/9] = [1.x] - [0.x] = 1 - 0 = 1 (not it)
B. 8 = [8/4] - [8/9] = [2] - [0.x] = 2 - 0 = 2 (this is it) the minimum value of n
C. 9 = 1
D. 12 = 3 - 1 = 2
E. 19 = 4 - 2 = 2

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This can simply be solved by plugging in values,

when u try 7: [7/4] is less than 2 and hence even without solving second term, we can discard this answee

try 8: [8/4] - [8/9] = 2 - [0. something] = 2- 0 (greatest integer less than 0.x is 0)
similarly all the other options work too, but the minimum possible value is 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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B. 8

Test the alternatives from smallest to tallest, to get the minumum possible.
[7/4] - [7/9] = 1 - 0 = 1 -> A. WRONG
[8/4] - [8/9] = 2 - 0 = 2 -> B. RIGHT

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
for the GMAT Club Olympics Competition

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