If n and t are positive integers, is n a factor of t ?

You can find that by trial and error: n=1 and n=2 does not satisfy n = 3^(n-2), but n=3 does. Now, if n>3 (4, 5, 6, ...), then RHS is always greater than LHS, so n=3 is the only solution.

You can solve this problem without finding the value of n in (1):

\(n = 3^{n-2}\) --> \(n=\frac{3^n}{9}\) --> \(9n=3^n\)

For (1)+(2): since from (2) \(t = 3^n\), then \(t=9n\), hence n is a factor of t.

Hope it' helps.
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If n and t are positive integers, is n a factor of t ?

(1) n = 3^(n-2) --> n=3 (only integer solution for this equation), but we know nothing about t, so this statement is not sufficient.

(2) t = 3^n --> if n=1 then the answer will be YES but if n=2 then t=9 and the answer will be NO. Not sufficient.

(1)+(2) As n=3 then t=3^n=27 and the answer to the question will be YES as 3 is a factor of 27. Sufficient.

Answer: C.
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hello Bunuel, How'd you figure it out in statement 1 that n=3. i do comprehend that n must be 3 but i cant figure it out by doing algebra.

please help

Hi All,

This question can be solved with a combination of arithmetic and TESTing VALUES.

We're told that N and T are POSITIVE INTEGERS. We're asked if N is a factor of T. This is a YES/NO question.

Fact 1: N = 3^(N−2)

Since this Fact tells us NOTHING about T, it's clearly insufficient. We can find the value of N without too much trouble though since we already know that it's a positive integer. With a little "brute force", we can find that N = 3 is the solution.
Fact 1 is INSUFFICIENT

Fact 2: T = 3^N

IF....
N = 1
T = 3
1 IS a factor of 3 so the answer to the question is YES

IF....
N = 2
T = 9
2 is NOT a factor of 9 so the answer to the question is NO
Fact 2 is INSUFFICIENT

Combined, we know...
N = 3
T = 3^N = 3^3 = 27
3 IS a factor of 27 so the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Quick Algebra question for Statement 1&2 combined:

If I plug in n = 3^(n-2) into t = 3^n I get:
n = 3^(3^(n-2))

when I rewrite it I eventually come to 3^3n * 1/3^6 = t

However this does not help me in any way... Where am i going wrong?

The highlighted steps are out of Sink

\(a^{(b^c)}\) is NOT equal to \(a^b*a^c\)

Whereas, \((a^b)^c\) = \(a^b*a^c\)

i.e. \(3^{(3^{(n-2)})}\) is NOT same as \(3^{3n} * 1/3^6\)

Thank you, so the only way would be to plug in values for n for \(3^{(3^{(n-2)})}\) ?

Or is there any way to rewrite this?

There are three ways

1) Plug-in the Values from Options
2) Take Logarithm on both sides and then solve further
3) Change the method and follow the methods given in other explanations

Third seems the Best to me

I hope it Helps!

We need to determine whether t/n = integer

Statement One Alone:

n = 3^(n - 2)

Since we do not have any information regarding t, statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

t = 3^n

We can substitute some numbers for n. For example, if n = 1, then t = 3^1 = 3 and 1 is a factor of 3. However, if n = 2, then t = 3^2 = 9 but 2 is not a factor of 9. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two, we can substitute 3^(n - 2) for n and 3^n for t in our question: t/n = integer ?

(3^n)/3^(n - 2) = integer ?

Since we are dividing similar bases, we can subtract the exponents and keep the base. Then we have:

3^(n - n + 2) = integer ?

3^2 = integer ?

9 = integer ?

Since 9 IS an integer. We have answered “yes” to the question.

Answer: C
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If n and t are positive integers, is n a factor of t ?


(1) \(n = 3^{n-2}\)

By plugging in numbers we see that only n = 3 works. However, we have no information on t. Insufficient.

(2) \(t = 3^n\)
\(t = 3^n\)
\(3 = 3^1 \)-- Yes
\(9 = 3^2 \)-- No
\(27 = 3^3\) -- Yes

We can't determine this without knowing what n is. Insufficient.

(1&2) This tells us that 27 = 3^3. Therefore, n is a factor of t. Sufficient.

Answer is C.
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Hi Bunuel chetan2u VeritasKarishma GMATBusters nick1816

Is the following solution correct?

(1) and (2) independently are not sufficient.

Combining,
For n to be a factor of t, t/n should give us an integer.

Given, n = 3^n-2 = 3^n/3^2 and t = 3^n

On dividing, t/n we get 3^2, which is an integer. Hence n is a factor of t. (C) sufficient.

Thanks in advance

Yes, you are absolutely correct.
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Asked: If n and t are positive integers, is n a factor of t ?


(1) \(n = 3^{n-2}\)
n = 3
NOT SUFFICIENT

(2) \(t = 3^n\)
There are multiple solutions for n & t
NOT SUFFICIENT

(1) + (2)
(1) \(n = 3^{n-2}\)
(2) \(t = 3^n\)
n = 3
t = 9
n=3 is a factor of t=9
SUFFICIENT

IMO C

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