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

Question

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

(2)  t = 3^n

Bunuel · Accepted Answer

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.

Bunuel · Answer

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.

alchemist009 · Answer

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

EMPOWERgmatRichC · Answer

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:  C

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

noTh1ng · Answer

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?

GMATinsight · Answer

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

noTh1ng · Answer

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?

GMATinsight · Answer

There are three ways 1) Plug-in the Values from Options :roll:

2) Take Logarithm on both sides and then solve further :oops:

3) Change the method and follow the methods given in other explanations :lol:

Third seems the Best to me :-D

I hope it Helps!

ScottTargetTestPrep · Answer

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

Basshead · Answer

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.

Laksh47 · Answer

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

chetan2u · Answer

Yes, you are absolutely correct.

Kinshook · Answer

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

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