Exponents and factors

[LulaBrazilia]

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

1) n = 3^(n-2)

2) t = 3^n

thanks!

[Patrick_GMATFix]

(1) by itself tells us nothing about t, while (2) alone does not restrict what n could be so it can't be sufficient. Together the statements are a lot more interesting. I go through the question in detail in the full solution below (taken from the GMATFix App).



-Patrick

[Brent@GMATPrepNow]

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?

Statement 1: n = 3^(n-2)
IMPORTANT: Since we are told nothing about t, we might automatically conclude that statement 1 is not sufficient. However, if n = 1, then this statement would, indeed, be sufficient since 1 would have to be a factor of t. So, before we can conclude that this statement is not sufficient, we must first ensure that n does not equal zero.
To do so, we'll plug n = 1 into the given equation to see if it works.
So, does 1 = 3^(1-2)? No, it does not. So, n ≠ 1.
Since n ≠ 1, we cannot determine whether or not n is a factor of t
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: t = 3^n
There are several values of n and t that satisfy this condition. Here are two:
Case a: n = 1 and t = 3 in which case n is a factor of t
Case b: n = 2 and t = 9 in which case n is NOT a factor of t
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that n = 3^(n-2)
Statement 2 tells us that t = 3^n
IMPORTANT: We want to determine whether n is a factor of t. If n is a factor of t then t/n will be an integer. So, let's take the 2nd equation divide it by the 1st equation.
When we do this, we get: t/n = [3^n]/[3^(n-2)]
Simplify: t/n = 3^2 [after applying the Quotient Law for exponents]
Evaluate: t/n = 9
Since t/n is an INTEGER, we know that t is divisible by n.
In other words, n is a factor of t
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

[GMATGuruNY]

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

1) n = 3^(n-2)

2) t = 3^n

thanks!

Statement 1: n = 3^(n-2)
If n=1, then 3^(n-2) = 3¯¹ = 1/3.
If n=2, then 3^(n-2) = 3� = 1.
If n=3, then 3^(n-2) = 3¹ = 3.
If n=4, then 3^(n-2) = 3² = 9.

Only n=3 satisfies the constraint that n = 3^(n-2).
No information about t.
INSUFFICIENT.

Statement 2: t = 3^n
First, test the only value that satisfies statement 1.
Case 1: n=3
Here, t = 3³ = 27.
In this case -- since 3 is a factor of 27 -- n is factor of t.

Now test a different value for n.
Case 2: n=2
Here, t = 3² = 9.
In this case -- since 2 is NOT a factor of 9 -- n is NOT a factor of t.
INSUFFICIENT.

Statements combined:
Only Case 1 -- n=3, t=27 -- satisfies both statements.
Thus, n is a factor of t.
SUFFICIENT.

The correct answer is C.