If $m$ and $n$ are natural numbers such that $2^m - 2^n = 960$, what is the value of $m$?

### Concept & Logic When dealing with the difference of two powers with the same base, factoring out the smaller power reveals a manageable integer equation. This transforms an exponential problem into a prime factorization problem. $$ a^x - a^y = a^y(a^{x-y} - 1) $$ ### Step-by-Step Solution * **Given:** $2^m - 2^n = 960$ $m$ and $n$ are natural numbers. * **Calculation / Deduction:** Since the difference is positive ($960$), $m$ must be strictly greater than $n$. Factor out the smaller power, $2^n$, from the left side of the equation: $$ 2^n(2^{m-n} - 1) = 960 $$ Now, break down $960$ into its prime factors, separating the power of $2$ from the odd integer part. $960 = 32 \times 30$ $960 = 64 \times 15$ $$ 960 = 2^6 \times 15 $$ Substitute this back into the equation: $$ 2^n(2^{m-n} - 1) = 2^6 \times 15 $$ By comparing the even (power of 2) and odd components on both sides: 1. The power of $2$: $$ 2^n = 2^6 \implies n = 6 $$ 2. The odd factor: $$ 2^{m-n} - 1 = 15 $$ $$ 2^{m-6} - 1 = 15 $$ $$ 2^{m-6} = 16 $$ $$ 2^{m-6} = 2^4 $$ Equating the exponents: $$ m - 6 = 4 \implies m = 10 $$ ### Exam Strategy & Shortcut **Option Substitution:** In competitive exams, directly testing the options in the equation $2^m - 2^n = 960$ is incredibly fast. * Try Option (a) $m = 10$: $2^{10} = 1024$. The equation becomes $1024 - 2^n = 960$. $2^n = 1024 - 960 = 64$. Since $64$ is exactly $2^6$, $n=6$ is a valid natural number. Option (a) fits perfectly. ### Common Pitfall A common mistake is trying to apply logarithms immediately to solve $2^m - 2^n = 960$, which leads to a mathematical dead-end because $\log(A - B)$ does not simplify. Factoring must come first. ### Final Answer **Therefore, the correct answer is 10.**