LulaBrazilia wrote: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
