Data Sufficiency

If \(2^x\cdot 7^y=z,\) what is the value of \(z?\)

(1) \(x-y=1\)

(2) \(2^x=8\)

Answer: C

Source: Princeton Review

$$Target\ qestion:\ 2^x\cdot7^y=z$$
Statement 1: x - y = 1
Only 2 things can be deduced from the information provided.
- x > y
- x and y are 2 consecutive integers.
The exact value of x and y is unknown, and as such, the target question cannot be answered. So, therefore, statement 2 is NOT SUFFICIENT.
$$Statement\ 2:\ 2^x=8$$
$$2^x=2^3$$
$$x==$$
Since the value of y is unknown, the target question remain unsolvable. Hence, statement 2 is NOT SUFFICIENT.

Combining both statements together:
From statement 1: x - y = 1
From statement 2: x = 3
Substituting the value of x in statement 2 into statement 1
3 - y = 1
3 - 1 = y
y = 2
x = 3 and y = 2
$$Therefore,\ 2^3\cdot7^2=8\cdot49=392$$
The value of z on combining both statements is 392. Therefore, this condition is SUFFICIENT. Hence, Option C is the correct answer.