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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 14:57
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Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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f(n)= l 3^n- 3^ (3x15) - 3^ (4x2) l
= l 3^n - 3^45 -3^8 l
the minimum value can be acchieved if we remove the biggest value inside the modulus i.e -3^45...
so n= 45
Ans=c
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 19:13
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ipon simplifying the above equation will look like this
|3 ^ n - 3 ^ 45 - 3^8|

so for least value of f(n)use max value of n that can be 45 and above values but if you go forvalue above 45 the in those cases distance between expression increase as modulus is there and values increases so best possible answer would be 45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 19:47
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answer c 45

absolute value of 3^n - 3^3 x ^15 equals 3^ 45 - 3^4 x ^2 equals 3^8

if 45 than
3^45 - 3^45 - 3^8


Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 19:59
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f(n) = |3^n-27^15-81^2|

f(n) = |3^n-3^45-3^8|

Now notice that from options if n = 15 value will be |3^15-3^45-3^8| , when n = 44 , this value will be smaller than previous value , when n=45 this will be smallest value which is 3^8. when n = 46 value again starts increasing. When n = 49 , It will have more value.

So at n =45 , f(n) is having smallest value which is 3^8. This is solved by POE.

So answer is C.
Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 19:59
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This expression gives |3^n - 3^45 - 3^8| clearly it would be NIL when n = 45.
Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 21:47
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\(f(n) = |3^{n} - 3^{45} - 3^{8}|\)

now if n = 45,
\(f(n) = |3^{8}|\)

if n=46,
\(f(n) = |(3^{45} * 3) - 3^{45} - 3^8|\)
\(f(n) = |2*3^{45} - 3^{8}|\)
it will be way bigger than \(3^{8}\).

Any other options will be bigger for the same reason

ans: option C (45)
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 21:58
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Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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We can simplify this as: \(f(n)=|3^n-3^{45}-3^8|\)

We want that bracket to be as small as possible, so we're looking for values that negate \(3^{45}+3^8\) as much as possible. \(3^8\) is relatively small compared to \(3^{45}\).

We're probably contemplating 45 and 46 most, just by eyeballing the numbers. \(3^{46}\), after subtracting the other values, still leaves about 2/3 of its value. \(3^{45}\) on the other hand only leaves \(3^8\), which would be a relatively small number. So C is our answer.
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 22:38
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imo. C = 45
F(n)=|3^n - 3^45 - 3^8| ; extracting 3^8 since it is smaller
= |3^n -3^8(3^37 + 1) | ; 3^37+1 is equivalent to just 3^37
so |3^n - 3^45| ; i.e. n = 45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 22:45
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f(n) = |3^n - 3^45 - 3^8| = |3^8*(3^(n-8) - 3^37 - 1)|
To minimize f(n), 3^(n-8) = 3^37 => n = 45
Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 22:51
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Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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As all of the three numbers are odd, the value of f(n) cannot be zero but can be close to zero

In that case,

3^n = 27^15 + 81^2

3^n = 3^45 + 3^8

As 3^8 is a smaller number in comparision to 3^45, we can ignore that.

Hence, 3^n is approximately equal to 3^45

Option C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 23 Dec 2025, 23:25
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Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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f(n) = mod (3^n -27^15 -81^2)

Smallest possible value of f(n) occurs at which value of n?

f(n) = mod ( 3^n - 3^45 - 3^8)

At n =44, we get f(n) = mod (-2*3^44 -3^8) = 2*3^44 + 3^8

At n =45, we get f(n) = mod ( 3^45 - 3^45 - 3^8) = mod ( -3^8) = 3^8

At n=46 , we get f(n) = mod(2*3^45 - 3^8) = 2*3^45 + 3^8

So, with increasing n , we are going to get an output of (something + 3^8 ) >>> 3^8 ,which occurs @ n=45

n = 45

Option C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 01:03
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The function can be rewritten as f(n)= I3^n-3^45-3^8I
Looking at 3^8 compared to 3^45 is quite insignificant so we can ignore it and just focus on getting a number that is closest to 3^45
In this case the value of n that gives a value closest to 3^45-3^8 = 45
Looking at 46 gives 3.3^45 which is actually tripple 3^45 so the answer is 45
Ans C
Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 01:14
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f(n) = |3^n - 27^15 - 81^2|, f(n) min
= |3^n - 3^45 - 3^8| >= 3^8, "=" occurs when 3^n = 3^45 => n = 45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 01:53
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The answer is 45. - Option C. I broke down f(n) = |3^n - (3^3)^15 - (3^4)^2| = |3^n - 3^45-3^8| if n were 45 then it would leave me with 3^8 which is the least value possible out of all options.
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 02:22
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Simplifying f(n) =
\(|3^n - 27^{15} - 81^2|\\
|3^n - 3^{45} - 3^8|\)

If we notice and take n = 45, we get f(n) as \(3^8\), which is the least.
Because for every other value of n, we get a higher power for 3 and a higher value of f(n).
For example, n = 15
\(|3^{15} - 3^{45} - 3^8|\\
|3^8(3^7 - 3^{37} - 1)|\)
We can see here that f(n) will be greater than \(3^8\) (f(n) is the absolute value, so even though inside the modulus it is negative, it doesn't matter).

Option C.
Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 02:38
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I'm glad this is the second question that concerns the powers of 3, because solving the first one does help prepare some calculations for this one :)

I used to be square of algebra. Wait - I am still scared of algebra, just not as much as I was. I learned, a lot of algebra hides something rather simple on the inside.

And in this case, it's the fact that EVERYTHING can simplified to the powers of 3. So before we go on to the options, let's do that for this big bad equation.

3^n - We're good here.
27^15 = 3^15*3^15*3^15 = 3^45
81^2 = 3^2*3^2*3^2*3*2 =3^8.

So, we just need the smallest value when we substituted the function (n) with a number while solving |3^n - 3^45 - 3^8|

Now, what else can we see here that the equation tells us clearly? That we take absolute values - | | - right?

This just means, the further down we go into the negatives, the larger the value will get. And, naturally, the further up we go into the positives, the larger the value. We just need a value that's the closest distance to 0 on a number line.

Now, we can check out the options.

A: This is the smallest of the powers, but that's exactly why this is not the choice. If we deduct exponentially higher powers of 3 - 3^45 and 3^8 - from 3^15, we'll go deep in the negative territory.

B: So, we can first look at 3^44 - 3^45. This will take us, although to a greater degree, well down the negatives, as that just one difference of powers at the ^45 and ^45 is in trillions, if not quadrillions. So, clearly, this won't work.

For the same reasons, we can eliminate D & E. Try however you want, the values to these will be massive.

C, clearly, is the right answer in how, when we deduct 3^45 with 3^45, we're at a solid 0, from which 3^8, which is really nothing more than a 4-digit number, will deduct, and lead to answer that's exponentially smaller than the others.


Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 03:08
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Bunuel
If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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we need least absolute value of f(n) because the equation is under mod so no negative values.

Let's simplify the equation

| 3^n - ((3)^ 3)15 - ((3)^4)2 |
= | 3^n - 3 ^45 - 3^8|

let's test values now

A - 15 --- No, if there was no mod, we could have considered it because it will give big absolute value, but now we need smallest absolute number so No

B - 44 --- 3^(44) - 3^45 -3^8 = 3^44(1 - 3) -3^8 = |-2* 3^44 - 3 ^ 8| ---- ok better than first option

C - 45 --- ok so two number will directly cancel and we will be left with = |-(3^8) | --- Lowest value till now

D - 46 --- it's absolute value will not be less than C | 2* 3^ 45 - 3^8|

E - 49 ---- value will be more than D so no need to check

Our answer is C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 03:11
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f(n) = | 3^n - 27^15 - 81^2|
= |3^n - 3^(3*15) - 3^(4*2)|
= |3^n - 3^45 - 3^8|

Given 3^45 >> 3^8, for n = 45, we get the smallest value of f(n) = 3^8.
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 03:13
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On simplifying the equation, we get

f(n) = |3^n - 3^45 - 3^8|

Then I substitute n with each of the options. All of them give values with the factor of 3^8 which makes them easier to compare. The smallest value is for option C which gives just 3^8.

Hence (C)
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 03:47
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If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49


Fn cannot be negative the list of value can be zero,

For fn to be zero, let's try to make positive side as close as possible with negative -

LHS = 27^15 + 81^2 = 3^45 + 3^8

RHS = 3^n is closest to RHS when n is 45. Answer C. 45
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