VJesus12 wrote:If n and t are positive integers, is n a factor of t? $$\left(1\right)\ \ \ n=3^{n-2}.$$ $$\left(2\right)\ \ \ t=3^n.$$ The OA is the option C.
I've got confused here. Can any expert give me some help? Please.
We have n and t are positive integers; we have to determine whether n is a factor of t.
Let's analyze each statement one by one.
(1) n = 3^(n - 2)
We have no information about t. Insufficient.
(2) t = 3^n
Case 1: Say n = 1, then t = 3^n => t = 3^1 = 3. We see that n (= 1) is a factor of t(= 3). The answer is Yes.
Case 2: Say n = 2, then t = 3^n => t = 3^2 = 9. We see that n (= 2) is NOT a factor of t(= 9). The answer is No. Insufficient.
(1) and (2) combined
From (1), we have n = 3^(n - 2) => n = (3^n) / 9
From (2), we have t = 3^n . Replcaing the value of 3^n in n = (3^n) / 9, we get n = t/9.
Since n and t are positive integers, for n to be a positive integer, t must be a multiple of 9.
Say t = 9k, where k is any positive integer. Thus, n = (9k)/9 = k. Thus, k (= n) is a factor of 9k (= t). Sufficient.
The correct answer:
C
Hope this helps!
-Jay
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