GMAT Practice Question: If r and s are positive integers such that (2^{r})(4^{s})=16, then 2 r+s=
Question
If r and s are positive integers such that (2^{r})(4^{s})=16, then 2 r+s=
- 2
- 3
- 4
- 5
- 6
Topics: exponents, equations of primes
Solution
Step 1
We rewrite 4 as a power of 2 and note that 16 is a power of 2.
4 = 2^2 16 = 2^4
Step 2
We express the left-hand side entirely as a power of 2.
(2^r)(4^s) = 2^r(2^2)^s = 2^{r+2s} 2^{r+2s} = 2^4
Step 3
We match the exponents to get a linear equation and find the positive integer solutions, then compute the desired expression.
r + 2s = 4 s = 1, r = 2 2r + s = 2 \times 2 + 1 = 5
Answer
5
