What is the smallest integer k such that 3^(1 - k)/3000

Question

What is the smallest integer k such that  \frac{3^{1-k}}{3000}<1?

A) -2

B) -3

C) -4

D) -5

E) -6

modiassumenda · Answer

We need:
3^(1−k) / 3000 < 1
Multiply both sides by 3000:
3^(1−k) < 3000
Now check powers of 3 near 3000:
3^7 = 2187
3^8 = 6561
Since 2187 < 3000 < 6561, we need:
1 − k < 8
⇒ −k < 7
⇒ k > −7
The smallest integer greater than −7 is:
k = −6
Check:
3^(1−(−6)) / 3000 = 3^7 / 3000 = 2187 / 3000 < 1
Therefore, the smallest integer k is  -6 .
 E. -6

AdarshSambare · Answer

We need:
3^(1−k) / 3000 < 1
Multiply both sides by 3000:
3^(1−k) < 3000
Now check powers of 3 near 3000:
3^7 = 2187
3^8 = 6561
Since 2187 < 3000 < 6561, we need:
1 − k < 8
⇒ −k < 7
⇒ k > −7
The smallest integer greater than −7 is:
k = −6
Check:
3^(1−(−6)) / 3000 = 3^7 / 3000 = 2187 / 3000 < 1
Therefore, the smallest integer k is  E. -6.

dotsoftme · Answer

To find the smallest integer $k$ that satisfies the inequality:
$$\frac{3^{1-k}}{3000} < 1$$
We can solve it step-by-step:
 Step 1: Simplify the inequality 
Multiply both sides by $3000$:
$$3^{1-k} < 3000$$
 Step 2: Understand the relationship with $k$ 
We are looking for the  smallest  integer $k$. Because $k$ has a negative sign in the exponent ($1-k$), making $k$ smaller will make the exponent $1-k$ larger, which in turn increases the value of $3^{1-k}$. Therefore, the smallest possible value of $k$ will give us the largest power of $3$ that remains strictly less than $3000$.
 Step 3: Test the powers of 3 
Let's find the powers of $3$ closest to $3000$:
 
 $3^5 = 243$ 
 $3^6 = 729$ 
 $3^7 = 2187$ (This is less than 3000) 
 $3^8 = 6561$ (This exceeds 3000)  
Thus, the maximum integer value that the exponent $(1-k)$ can take is $7$.
 Step 4: Solve for $k$ 
$$1 - k \le 7$$
$$1 - 7 \le k$$
$$-6 \le k$$
Since $k$ must be greater than or equal to $-6$, the smallest integer value $k$ can take is  $-6$ .
 Correct Answer:E) -6

onlyPlanA · Answer

Although I applied the above method as well to solve this, isn't this question somewhat confusing? If you substitute all the options in place of k, all those options fit. For example,

Since:
3^(1-k) < 3000

Then substituting Option-A: -2 in place of k,
 3^1-(-2) = 3^3 = 27, which is less than 3000. 
Same goes for other options. So, shouldn't A be the answer being smallest integer? Can someone please enlighten me where I am wrong?

suntprovident · Answer

Bunuel  can you help why -2 is incorrect ?

architkap · Answer

This is what I think of the question -

It's asking us the smallest value of K where the fraction is less than 1, its not asking us the smallest fraction possible for a value of K. When K is -6 the fraction is still less than 1. when K is -7 it is over 1.

Hope that helps.

Daphnee · Answer

All the options fit but -6 is smaller than -2, and also is the smallest among all the options so that's why the answer is E.
Remember 6 > 2 but -6 < -2

onlyPlanA · Answer

Yes, thank you. Only after few hours of posting this reply, I realized my blunder that -6 < -2. But anyway, thanks!

What is the smallest integer k such that 3^(1 - k)/3000 < 1?

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