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

[melguy]

Hello

Please help me with this question.

Thanks

[crackverbal]

Hi Melguy,

The best way to solve this question is to plug in values. Keep in mind that this is a YES/NO DS and for n to be a factor of t, n should perfectly divide t or in other words t/n has to be an integer.

Statement 1 : n = 3^(n-2)

n = 3^(n-2) -----> n = (3^n)/(3^2) -----> 9n = 3^n

The only value of n that satisfies the above equation is n = 3. Statement gives us the value of n but does not give us any information about t. Insufficient.

Statement 2 : t = 3^n

Plugging in n = 1 we get t = 3, this gives us a YES, since t/n is an integer.
Plugging in n = 2 we get t = 9, this gives us a NO, since t/n is not an integer. Insufficient.

Combining statements 1 and 2

The only value of n is 3, so the only value of t is 27, so t/n will always be an integer. Sufficient.

OA : C

Learn how to plug in smart values to solve DS questions and some more strategies to solve PS and DS questions here https://gmatonline.crackverbal.com/cours ... nt-on-gmat

CrackVerbal Academics Team

[ceilidh.erickson]

This question can also be solved CONCEPTUALLY, using properties of exponents:

If n and t are positive integers, is n a factor of t?
(1) n = 3^(n - 2)
(2) t = 3^n

Target question: is n a factor of t? Or in other words, does t/n = an integer?

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

This tells us nothing about t, so it can't possibly be sufficient to answer the question.

(2) t = 3^n

This tells us that t is a power of 3, greater than or equal to 3^1. However, a given (positive) power of a number does not have to be divisible by the exponent itself; it only needs to be divisible by the base. For example, in 3^2 = 9, 9 is not divisible by 2, but it is divisible by 3.

Insufficient.

(1 & 2) Together

If t = 3^n and n = 3^(n - 2), then t/n = (3^n)/(3^(n-2)).
When we simplify, we get 3^2. We get an integer result when we divide t/n, so n must be factor of t. Sufficient.

The answer is C.

[Scott@TargetTestPrep]

If n and t are positive integers, is n a factor of t?
(1) n = 3^(n - 2)
(2) t = 3^n

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

[[email protected]]

Hi melguy,

This question can be solved by TESTing VALUES and a bit of 'brute force'

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.

1) N = 3^(N-2)

This Fact tells us nothing about T, so it's clearly insufficient, but the equation is rather specific, so there can't be that many potential solutions. Let's see if we can quickly 'brute force' the solution by TESTing VALUES until we have enough information to stop working...

IF...
N=1, then does 1 = 3^(-1)? No, so N CANNOT be 1
N=2, then does 2 = 3^(0)? No, so N CANNOT be 2
N=3, then does 3 = 3^(1)? YES, so N CAN be 3
N=4, then does 4 = 3^(2)? No, so N CANNOT be 4

At this point, we can stop working because 3^(N-2) will get much larger as N increases than "N" does. We now know that the ONLY positive integer solution to this equation is N=3.
Fact 1 is INSUFFICIENT

2) T = 3^N

IF...
N=1, T=3, then the answer to the question is YES.
N=2, T=9, then the answer to the question is NO.
Fact 2 is INSUFFICIENT

Combined, we know that there's just one potential solution that fits both Facts:
N=3, T=27, then the answer to the question is YES.
Combined, SUFFICIENT

Final Answer: C

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