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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 04:17
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IMO C

N=45

We have to see that the answer will always be positive, so we need to bring f(n) to the smallest number.

when we see the eqn we notice everything can be re written with base 3.

So

f(n)=∣3^n−3^45−3^8|

If we put n=15, it will results in much larger solution, smallest solution will be somewhere near n=45 or 46

if we put n=45, result f(n)= 3^8
but if we put n=46, result f(n)= 2.3^45-3^8= much larger than 3^8
therefore n=45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 04:20
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powers of 27 and 81 with 3
27^15=(3^3)^15=3^45
81^2=(3$4)^2=3^8
f(n)=13^n-(3^45+3^8)1
3^45 3^8
3^45, (3^45+3^8)-3^45=3^8
3^46, 3^46-(3^45+3^8)=2x3^45-3^8
3^44, (3^45+3^8)-3^44=2x3^44+3^8
close=3^45 so n=45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 04:27
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f(n)=|3^n-27^15-81^2|
We first rewrite everything as powers of 3
27=3^3 so 27^15= 3^45
81=3^4 so 81^2= 3^8
so f(n)=|3^n-3^45-3^8|
Compare sizes; 3^45>3^8 so 3^45 is slightly larger than 3^45
Choose the closest power of 3 to minimize the absolute value 3^n
3^45: difference = 3^8 (small)
3^46: difference is 2.3^45-3^3 (huge)
clearly 3^45 is closer

The answer is n=45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 05:02
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We get the smallest value possible for an absolute number when we are as close as possible to 0.

Here we have a positive term 3^n and two negative terms: 3^45 and 3^8
so our goal is to get with 3^n as close as possible to the sum of the two negative terms.

In this case since 3^45 >>> 3^8 we have with n=45 a value of f(45)= 3^8

If n>45 instead we would get an error much bigger.

IMO C!
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 05:49
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\(f(n)=|3^n-27^{15}-81^2|=|3^n-3^{45}-3^8|\)
For this function to be the smallest, we need to either get the smallest positive value, or the highest (closest to zero) negative value.

A. \(3^{15} - 3^{45} - 3^8 = 3^{15}*(1-3^{30}) - 3^8 = (-3^{30}+1)*3^{15} - 3^8 \) => a large negative minus 3^8

B. \(3^{44} - 3^{45}-3^8 = 3^{44}*(1-3) - 3^8 = -2*3^{44} - 3^8\) => a large negative minus 3^8

C. \(3^{45} - 3^{45}-3^8 = -3^8\) => simply minus 3^8

D. \(3^{46} - 3^{45} - 3^8 = 3^{45}(3-1) - 3^8 = 2*3^{45} - 3^8\) => a large positive minus 3^8

E. \(3^{49} - 3^{45} - 3^8 = 3^{45}(3^4-1)-3^8 = 80*3^{45} - 3^8\) => an even larger positive minus 3^8

From the above transformations, it's clear that \(|-3^8|\) will have the smallest absolute value; therefore, the 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, 05:58
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f(n) = |3^n - 27^15-81^2|. For minimizing, we need to bring 3^n as close as possible to (27^15+81^2) i.e. (3^45+3^8) as they are the positive and negative parts.
Now in cases where n<45 => we are left with options at least in the range of 3^45. In cases where n > 45, we are left with results in the range of at least 3^45 again.
Only option which suffices n = 45. Hence, C.
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 06:34
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We want to minimize:
f(n) = |3^n - 27^15 - 81^2|

We can also write the same in the powers of 3:
27^15 =(3^3 )^15 =3^45
81 = 3^4 ⇒ 81^2 = ( 3^4 )^2 = 3^8
So, f(n) = ∣ 3^n − 3^45 − 3^8 ∣ .

By factorizing:
3^45+3^8=3^8((3^37)+1)

Since 3^34 is significantly larger than 3^8. We can approximate that 3^45 + 3^8 will only be somewhat greater than 3^45
So the value of 3^n that minimizes the absolute difference should be closest to 3^45 + 3^8

Now using the answer options, checking values near 3^45 : n = 44: 3^44 is much smaller than 3^45 + 3^8
n = 45: 3^45 is very close, but less than 3^45 + 3^8
n = 46 : 3^46 = 3 x ( 3 )^45 = which is significantly larger.
Clearly, 3^45 < 3^45 + 3^8 < 3^46, and since 3^8 < 3^45 , the closest power should be 3^45.

So for me the smallest value of f(n) should occur when n = 45.
Answer is C.

It took me a lot of time to get this solution. Can someone suggest a better way please?

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, 06:52
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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)?

f(n)=|3^n -3^45 -3^8|
Since 3^45>>>>3^8 the value of 3^n that will be closest to (3^35 +3^8) will be n=45. Further we can check the options closest to n=45
f(45) = |3^45 -3^45 -3^8| = |-3^8| = 3^8

n=45 results in the smallest possible values for f(n)
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 06:57
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simplifying the equation f(n) = |3^n - 3^45 - 3^8|
if I substitute n = 45 -> f(n) = 3^8
if I try more n = 46 -> f(n) = |2*3^45 - 3^8| which is a lot more
if I try less n = 44 -> f(n) = |-2*3^44 - 3^8| which again due to mod is a lot more
hence n = 45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 07:00
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f(n)= |3^n - 27^15 - 81^2| simplifies to. f(n)= |3^n - 3^45 -3^8|

3^8 is very small compared to 3^45, we can write

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

Now, 3^n should be nearest to 3^45

A. 3^15 is very small
B. 1/3 of 3^45
C. exact term
D. 3. 3^45 --> very large
E. 3^49 very large

Hence. C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 07:04
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The answer is C 45 .
27 ^ 15 = (3^3)^15 = 3^45
81 ^ 2 = 3^8
thus for the answer to be lowest n must be 45 then
f(n) = |3^45 - 3^45 - 3^8|
= 3 ^ 8
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 07:05
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f(n) = |3^n-3^45-3^8|

3^n-3^45-3^8 = 0
3^n = 3^45 + 3^8

3^n = 3^8{3^(45-8) + 3^(8-8)}

3^n = 3^8 (3^37+1)

Since 3^37 is a multiple of 3, 3^37+1 is not a mutiple of 3 which means this expression cannot be equat to 0.

Analyze option choices to check which is closest, we find that f(n) is smallest 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 24 Dec 2025, 07:15
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We are given,
f(n) = |3^n - 27^15 - 81^2|
=> f(n) = |3^n - 3^45 - 3^8|

We need to know for which value of n is f(n) the smallest

=> f(n) = |3^n - (3^8)*(3^37 + 1)|

We see that the powers closest to 45 should be considered here among the options

=> For,
n = 44 we get,
f(44) = 2*3^44 + 3^8

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

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

Therefore, n=45 gives the smallest value

C. 45
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 07:20
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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) = |3^n - 27^15 - 81^2|,
f(n) = |3^n - 3^45 - 3^8|

to minimize this f(n) should come close to 0
that happens when 3^n = 3^45 + 3^8
3^8 is very small compared to 3^45

so n = 45 gives minimum value among given options
Correct Answer: C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 07:36
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f(n) = | 3^n - 27^15 - 81^2 |
f(n) = | 3^n - 3^45 - 3^8 |

Now 3^8 is much smaller than 3^45 hence we need to find values closer to 3^45

Using n = 45 we get
f(n) = | 3^45 - 3^45 - 3^8 |
=> f(n) = 3^8

Using n = 44 we get
f(n) = | 3^44 - 3^45 - 3^8 |
Taking approximation
f(n) = | 3^44 - 3^45 | = | 3^44(1 - 3) | = 2*3^44
Much larger than 3^8

Using n = 46 we get
f(n) = | 3^46 - 3^45 - 3^8 |
Taking approximation
f(n) = | 3^46 - 3^45 |
f(n) = | 3^45 (3 - 1) | = 2*3^45
Much larger than 3^8

for n = 15 the value would be much greater than 3^8
For n = 49 the value would be much greater than 3^8

The smallest value comes to 3^8 for n = 45

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, 07:41
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Rewriting in powers of 3,
f(n) = |3^n-3^45-3^8|
Now, 3^45 + 3^8 = 3^8(3^37+1)
For f(n) to be smallest, 3^n should be closest to 3^45+3^8

3^45 < 3^45+3^8 < 3^46
3^n should be closest to 3^45 for minimum f(n) as 3^45 + 3^8 is closer to 3^45 than 3^46
So 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 24 Dec 2025, 07:42
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|3^n - 27^15 - 81^2|
|3^n - (3^3)^15 - (3^4)^2|
|3^n - 3^45 - 3^8|

f(n) is always non negative.

3^8 is much smaller than 3^45, so:
(3^n - 3^45) is near zero when n = 45

f(45) = 3^8

IMO C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 08:02
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f(n) is always positive or 0.

f(n) = abs(3^n-27^15-81^2) = abs(3^n-3^(3*15)-3^(4*2)) = abs(3^n-3^45-3^8) = abs(3^n-(3^45+3^8))

We know that 3^45 is a very large number, while 3^8 is significantly smaller. Therefore, the value of (3^45+3^8) is approximately 3^45.

abs(3^n-3^45) is 0 if n=45 -> f(45) = 3^8 = 6561 is its smallest possible value (with n integer).

Answer C
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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2 24 Dec 2025, 08:18
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All are powers of 3.
f(n) = |3^n-27^15-81^2| = |3^n-(3^3)^15-(3^4)^2| = |3^n-3^45-3^8| is always >=0

We want 3^n to be as close as possible to (3^45+3^8) -> n must be close to 45

if n=45, f(45) = |3^45-3^45-3^8| = 3^8
if n=46, f(46) = |3^46-3^45-3^8| = |3*3^45-3^45-3^8| = |2*3^45-3^8| = 2*3^45-3^8 > 3^8
if n=44, f(44) = |3^44-3^45-3^8| = |3^44-3*3^44-3^8| = |-2*3^44-3^8| = 2*3^44+3^8 > 3^8

n=45

The 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, 08:36
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f(n) is the absolute value of:
3^n - 27^15 - 81^2 = 3^n - (3^3)^15 - (3^4)^2 = 3^n - 3^45 - 3^8

For the given values:
if n=15 then f(15) = |3^15 - 3^45 - 3^8| is similar to 3^45 -> huge number
if n=44 then f(44) = |3^44 - 3^45 - 3^8| is similar to 2*3^44 -> huge number
if n=45 then f(45) = |3^45 - 3^45 - 3^8| = 3^8
if n=46 then f(46) = |3^46 - 3^45 - 3^8| is similar to 2*3^45 -> huge number
if n=49 then f(49) = |3^49 - 3^45 - 3^8| is similar to 3^49 -> huge number

minimum value for n=45

The correct answer is C
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