Question

Tìm m và n là số nguyên dương sao cho :2^m-2ⁿ=512

Answer 1

\(2^m-2^n=512\)

\(\Rightarrow2^m-2^n=2^9\)

\(\Rightarrow m=10;n=9\)

Answer 2

\(2^m-2^n=512\Leftrightarrow2^m-2^n=2^9\Leftrightarrow2^m>2^n\Leftrightarrow m>n\)

\(TH1:m-n=1\)

\(\Rightarrow2^m-2^n=2^n\left(2^{m-n}+1\right)=2^9\Leftrightarrow2^n.\left(2-1\right)=2^9\)

\(\Leftrightarrow2^n=2^9\Leftrightarrow n=9\)\(\Rightarrow m=10\)

\(TH2:m-n>2\),\(2^n\left(2^{m-n}+1\right)=2^9\)

Vế trái có thừa số \(2^{m-n}+1\)lẻ (Vì m - n >2 nên \(2^{m-n}\)chẵn\(\Leftrightarrow2^{m-n}+1\)lẻ)

Vậy m = 10; n = 9